Detach, detach, detach! Make this your mantra! Let him write all the letters he wants, and make all the demands he wants... They are pretty powerless protests, when you think about it, unless you let them wind you up!
He is clearly a very clever boy - perfectly clever enough to realise the truth of "OK, don't study. Your choice. But if you don't do it now, the rest of your life is going be very much harder. And we won't be around to support you then".
I do understand what it's like, Tiger, and how infuriating it is - my DS1 behaved very like this the year he was 15. (Except he didn't take it quite as far, and I didn't rise to the bait quite as often). He under-achieved at GCSEs, and totally wasted Y12, and went off the tracks a bit... But now he's back on them, doing a course he wants to do, and talking about university. So don't think our DS is on his 'last chance', because he really isn't.
Can I double-check something? Am I right in understanding that he now has no phone, no phone charger, no kindle, no guitar, no access to the computer except for homework and no wifi? All because he hasn't studied as much as you want him to?
If so, I'm afraid you have painted yourselves into a bit of a corner. You have been very heavy-handed (IMO), and what you are seeing now is "Might as well be hanged for a sheep as a lamb" behaviour. He has nothing left to lose, and no motivation to behave. He might as well be rude and challenging, because why should he not be, and what else can you do? You yourself have recognised the 'logical' next step is the totally ridiculous "tying him to a tree in the garden". That's a pretty good sign you've gone too far. :(
It's easy to get into situations like this, and hard to get out of them again.
You can't easily give him back privileges while he's behaving badly, and he has no reason to co-operate while he has no privileges. As I said above, the idea is for you to give him control, rather than for you to lose it - but at the moment you have lost it, I'm afraid.
I think the only way out of this impasse is for you to pick a calm moment (and I agree with niceguy that car journeys are good) and start 'negotiations' with him. Personally, I would probably say something like "I've been thinking. We're fighting, but really that's silly, because actually we want the same thing here. We want you to do well in your GCSEs and you want to do well... Can we talk about how we can all make that happen?" and then take it from there...
I would expect a bit of blustering at first (after all, he's clearly angry too) and probably he'll try saying things like "You give me back all my stuff and then I'll talk"... The trick to having successful negotiations is going to be to stay detached, and keep focused on what you want out of the situation.
But be prepared to make some 'concessions'. Because the alternative is that you stay locked in this impossible conflict for a very long time.
I don't know if that makes sense. It's late now so I may not be explaining myself well. I'll have another read tomorrow, probably when I'm back from work... :)