The bike is now fixed!
It rides and sounds 10 times better than it did before I had a mechanic look at it. I got a full service, new oil filter, tank cleaned out (and unblocked), had some bolts replaced, cleaned the carbs and all of the usual full service stuff on top of that. Needs to go back for new brake pads soon and definitely a new throttle cable, it's not very responsive, but still very heavy to handle.
All in all, the repairs and whatnot cost me £155. Turns out there was a lot of water in the engine, which was the biggest cause of all my problems.
Oh, and after all that, I'm now confident that I'm using my choke properly (which I was before anyway, it turns out).

