Wow, didn't expect to come into this thread and see a pile of of posts from people who haven't got the faintest idea what they are talking about!
I'll leave it at that.
Okay, on to the AE86.
I have worked on and driven these cars in anger for years.
The AE86 (chassis code) known as the Corolla GT Coupe in the UK, Levin/Trueno/Sprinter/Apex etc in various other markets.
Forget the fact that it's a "Mid-80's Corolla" because it isn't really. Interior is horrible but if you are looking at an AE86 for the right reasons then you won't care.
All UK Models should have (to the best of my knowledge) a DOHC 16v 4AG engine as used in the Mk1 MR2. Between 120 and 130 bhp if in good fettle.
Every UK AE86 I have driven or worked on had a factory fit LSD although these can get very "soft" with age and end up behaving like an open differential.
If you are the sort of person who can see absolutely no reason why a road car would need an LSD or what you would gain from it on an enjoyment level other than a slight increase in cornering traction then you should probably turn around now and leave the thread because the whole concept of enjoying driving sideways is going to almost certainly be lost on you.
Drivetrain is generally very robust - conventional RWD setup, gearboxes can become whiney with age and halfshafts can break if youy are extremely cack handed on uneven surfaces.
Driving experience is extremely rewarding. Sharp steering, good turn-in, minimal understeer. Nicely balanced, easy (IMO) to drive fast and very very easy to encourage sideways action so long as your clutch and LSD are tight. Some people feel the AE86 lacks power as standard but this is normally because these people have stepped out of 2-300bhp Silvias. Engines can suffer head gasket failure and cam wear but nothing to particularly look out for. They are not habitually unreliable cars but you will need to consider the age of it and the associated wear and tear that happens over time.
The main thing you are looking for is rot. Everywhere. Particularly the A posts, sills, rear chassis rails, rear pockets (behind the bumper), spare wheel well, floorpan/bulkhead seam. All panels. Sunroof. Window surrounds (hard to repair neatly) Inner wings and engine bay in general don't seem to suffer so badly. UK cars are usually about a million times worse for rot than imported cars.
It's a shame that the car has become a target for big spenders and "scene tax" following it's popularity in Initial D and the drift bandwagon in general. Now people just harp on about the cost and forget that it's actually a very good drivers car. Or assume you are a drift king wannabe with more money than sense.
Would I buy one? No, not now. Too expensive for what it is. Would I HAVE one? Without a doubt yes. If prices started coming back to the 2k mark I'd consider one again.
Alternatives?
Nissan S12 has been mentioned, also fun to drive but much larger than the AE86, nowhere near as rewarding to drive, tends to be a bit "all or nothing" you are either wheelspinning on the spot or up against the lockstops whereas the AE86 is so much more progressive. The MX-5 is the closest thing I can compare it to really, has that same nimble feel to it. A good one (a set of Bilsteins or random Japanese coilovers of your choice is ideal replacement for the tired OE dampers) will be very rewarding to drive if that is the sort of car you like - and coming from an Integra I think it will be.
Lots of fat useless chumps buy AE86's and think that jumping in with their foot to the floor is just going to transform them into some kind of driving god. Of course it won't, and I think you know that.
My advice to you personally would be to find someone who has one and really knows how to drive it and get them to take you out down a single track country lane with the hammer down. Then have a go yourself. I guarantee you will be grinning from ear to ear when you get out of the car.
Hope that was more helpful than "LOL ITZ AN AYTIES COROLLA INNIT? SHED TBH"