S18 RSG

To answer your questions 2 and 3:
2) I'm almost certain that level of power is attainable with a 2.5" exhaust. Plenty of tests revolving around this subject on the internet, not just in GT86's. Most test seem to conclude that a 3" exhaust only really makes a big difference when you get into 500bhp+ territory. Sure, you'll probably see a few extra hp even around 300, but if you already have 2.5" system, there's little point in upgrading.
3) Supporting mods for a safe 300whp and a HKS charger would almost certainly include uprated internals (pistons and rods as a minimum). An uprated clutch, brakes, minimum 255 wide rear tyres (most recommend 265's for this power) and an oil cooler would be the starting point. Factor in £4.8k for HKS kit and fitting (Fensport prices), £3.5-4k for the uprated internals plus labour, £500-1k for clutch and labour and even if you ignore the parts not related to the engine, you're in for close to £10k already.
I also completely forgot you have the auto box, instead of manual, in which case, you're going to really struggle going anywhere above the norm. Learn from Keith's mistakes, as he spent a hell of a lot of time and money finding out what you can now know for free. In the end, I think your best bet would be to pick the kit you want (I would recommend TD's turbo for obvious reasons), install a 4.67FD to more closely match the manual box, then if you want some extra speed, add lightness rather than power.
As a final side note, don't necessarily pay attention to those who say a supercharger is "kinder to the engine" than a turbo. That's all to do with the map. I would actually argue that a turbo application can be tuned to be even kinder than a supercharger, as you have full control over boost pressure by adjusting the wastegate. A supercharger makes boost as the engine builds revs, and you adjust boost by fitting smaller pulleys/belts, so the boost is much less flexible than with a turbo. Just something to think about.

To answer your questions 2 and 3:
2) I'm almost certain that level of power is attainable with a 2.5" exhaust. Plenty of tests revolving around this subject on the internet, not just in GT86's. Most test seem to conclude that a 3" exhaust only really makes a big difference when you get into 500bhp+ territory. Sure, you'll probably see a few extra hp even around 300, but if you already have 2.5" system, there's little point in upgrading.
3) Supporting mods for a safe 300whp and a HKS charger would almost certainly include uprated internals (pistons and rods as a minimum). An uprated clutch, brakes, minimum 255 wide rear tyres (most recommend 265's for this power) and an oil cooler would be the starting point. Factor in £4.8k for HKS kit and fitting (Fensport prices), £3.5-4k for the uprated internals plus labour, £500-1k for clutch and labour and even if you ignore the parts not related to the engine, you're in for close to £10k already.
I also completely forgot you have the auto box, instead of manual, in which case, you're going to really struggle going anywhere above the norm. Learn from Keith's mistakes, as he spent a hell of a lot of time and money finding out what you can now know for free. In the end, I think your best bet would be to pick the kit you want (I would recommend TD's turbo for obvious reasons), install a 4.67FD to more closely match the manual box, then if you want some extra speed, add lightness rather than power.
As a final side note, don't necessarily pay attention to those who say a supercharger is "kinder to the engine" than a turbo. That's all to do with the map. I would actually argue that a turbo application can be tuned to be even kinder than a supercharger, as you have full control over boost pressure by adjusting the wastegate. A supercharger makes boost as the engine builds revs, and you adjust boost by fitting smaller pulleys/belts, so the boost is much less flexible than with a turbo. Just something to think about.

How about the childish things, like noise?
The TVS chargers give off a feint whine, but nothing like as loud as the Sprintex. The TD kit on the other hand is classic Supra/Skyline soundtrack. Open wastegate, flutter when lifting off etc. It's by no means subtle. I can't imagine it would be a deciding factor, but certainly something to consider I would think. 

If Cosworth is out of your budget, so is a safe 300whp.
You can achieve 300whp with the items you've mentioned, but without upgrading internals, you're at huge risk of blowing the engine, in which case, you're talking about doubling your investment.
I'm interested in why you think 300whp is necessary. Mike's stage 2 tune on his demo car is running 340 at the fly, and it's on the limit of what's safe on the stock engine thanks to his tuning, but to me, it's maybe a little too much for the road.
I would suggest 300hp at the crank is all you'd ever need on the road. Sure, on the track you can go higher still, but for 2-3 track days a year, it's not really a track car. 
There's a good reason most FI owners stay at around 280hp. It's safe on the engine, requires no supporting mods beyond off the shelf kits, and puts the car into Cayman S/M3 territory anyway.

I also wanted a TTRS before the GT86 (among other things anyway) but straight line speed, while fun, gets boring a lot quicker than fun in the corners. Admittedly, I now have both (although not quite TTRS fast yet), but I have never once regretted my decision.
On topic, I've wanted that head unit for months, unfortunately, brakes and tyres are required, a new head unit isn't, so it will be a few months before I'd be able to buy it

I just went on the internet and bought new pads. As long as they're better than stock (which they will be), does it really make that much difference? (This is a rhetocrical question)

I just went on the internet and bought new pads. As long as they're better than stock (which they will be), does it really make that much difference? (This is a rhetocrical question)

Ride quality is great. Springs are fairly soft for aftermarket coilovers (5kg/mm all around), which is stiffer than stock, but in my experience nowhere near stiff enough to feel uncomfortable.
They require a little bit of setup however, as the damper settings can be a little fiddly to work out, but once you've dialled them in, they're great. 
A few people who have experience of GT86's have commented that my setup is one of the best they've tried, most negative comments you will get about these will be from people who haven't tried them

It's pretty straight forward. Once you've removed the front plate, there are two screws holding the "brick" in place. Just undo them, then remove the screw thread clips and you're done.
I drilled holes in my plate in the same locations as the holes for the brick, and used slightly larger white bolts to hold it. Works well.

There's been a lot of discussion about tyres in the past few days, and judging by that evidence, you'll struggle to get one definitive answer.
It really depends on your budget.
A good, relatively cheap and well tested tyre is the Yoko V105 as Jeff mentioned, in a similar price range are the Hankook V12 Evo 2 and Vredestein Sessanta's, both of which have great test results with magazines/online articles.
On the more premium end, you have the Michelin PS4, Continental Conti6 and Goodyear Eagle F1 Asy 3. 
All those mentioned above will be significantly better than what you have without doubt, and should have no problem handling all weather conditions.

You gotta remember though, tyre sizes aren't just about handling. Running a 225 with nearly 50% more power than stock means I can barely enjoy the power, as it's spinning up 1st through 3rd (sometimes even 4th in damper conditions) when I get on it.
Going to a 235, I would say, is hardly gonna make a difference. In fact, I know Mark is running 235's and I'm sure he said he had some traction issues at that size.
A 245 I feel would be ideal. The problem is, a set of PS4's in 255/35/18 are cheaper than a fair amount of tyres at 245 (only Yoko V105's and Vredestein Sessanta's are cheaper, of those I would run anyway), perhaps due to the more common size, so while it may be a little wider of a tyre than I had planned, I feel it's justified.
Also, 255/35/18 and 255/40/18 are what BMW, Mercedes and Audi all run on OEM 8.5J wheels. Now I'm not saying that's all the proof we need, but it shows that it's clearly a fine setup.

May the spirit of Daigo guide you 🙏

Malvern Tyres (not sure if they're Nationwide, or just Wales) can do them for £135 fitted. Gonna pop by on my way home and see if they can't knock that down to £500 for a set of 4 (£125 each). If so, I'll probably get a set on order.

Mine are 35 offset front, 30 rear, but obviously half an inch narrower, so we're probably in a similar situation.

I don't believe so. Price is here: https://www.camskill.co.uk/m61b0s22p135329/Michelin_Tyres_Car_Michelin_Pilot_Sport_4_PS4_-_255_35_R18_(94Y)_XL_TL_Fuel_Eff_%3A_C_Wet_Grip%3A_A_NoiseClass%3A_2_Noise%3A_71dB
However, fitting is never a concern for myself as a good friend of mine is the manager of a local tyre place. Not so much of a friend that he can get me great deals on tyre prices, but he always fits tyres for free, or very little cost.

Another consideration here. @Riceburner informed me that I should be ok with a 255/35 on an 8.5 (something I've noticed is commonplace on the BMW forums), and the Michelin PS4's are only £130 each at that size, £20 per corner cheaper than 245/35's!
Gotta be a good option right?

Well, as there are clearly still some questions, I'll try and take some time to do some testing while I'm in the car.
Step 1: Sound deadening. This is the cheapest and easiest mod, and realistically should make a noticeable difference.
Step 2: Fit the amp with stock speakers.
Step 3: Swap out the speakers for the new Focal items.
To answer a question I glossed over, yes, the head unit will end up getting changed, just need to find something OEM looking with good navigation software, DAB etc. and I'll pick one up. As I said initially, this is just something that's been bugging me since I bought the car, and shouldn't be too expensive or difficult to improve.
At the end of the day, a car will never be a match for listening to music at home, and that's why I've invested well into four figures on my home audio setup, but for the times I am listening to music in the car, I'd like to at least enjoy it, not have some distorted, completely flat version of my favourite songs

I always think about the same story when talking about this kinda thing. A few years ago my cousin bought an "eBay special" chip (a resistor in a box) that promised 15% power or something on his Corsa diesel. He said it was significantly faster after he fitted it. My dad told him he could make it even faster, and would tweak it. My dad took this device off his car and gave it back to my cousin. He said it felt even better after my dad messed with it.
Placebo effect is an amazing thing, and noise is the biggest cause of it. I'm not saying you're that stupid btw  Just a funny story. I've had similar feelings when I've made mods, it's only after removing them did I realise how little, if any, differences were made.

I usually find a brick wall does a decent job, and a fraction of the price.

Official one has a drill template, but I think 90% of people just stick it on. But for the price difference, it's not a huge problem.

12x6

Seems like good work and good research done.
Funny enough, my car is almost silent with no cats and no silencers. My exhaust guy fitted his own design backbox to quieten mine down, and of course the turbo helps quieten it even more. Without the screamer pipe I wouldn't be surprised if it's quieter than stock.
Interesting consideration for people stuck between supercharging and turbocharging. The lengths at which you have to go to achieve what you can get for free in a turbo setup.

Only other colour I can use is blue, as the intercooler piping is blue, but if I do that I'll have to change wheel nuts and all sorts of other things to blue as well. Aesthetics Paul. Think of the children.

Short video taken today by @Jared or his missus. Finally made me realise how mental the screamer pipe actually is.
Thought it might be of some interest to others  
 
 

I hope they're black. If I pretend I'm an adult for a second, I really can't afford it this month, but if they're red I'd probably forget I'm an adult...