[Daisrocks]

Damn, that sounds like a tricky problem. I'll try and answer some of your questions.
1) I have the Haynes manual for that engine but I am not selling it. You can buy it by choosing the manual for the 86-89 Celica. I'll have a look at it when I get home and if it has any details on the problem I'll let you know. And yes, it has information on the vacuum system/hoses/sensors.
2) As far as I know the difference between the 3S-GE and the 3S-GTE is that the GTE has a turbo fitted. So you'll have all the normal differences that that entails: lower compression cylinder, wiring loom, air induction pipe work, etc...
3) Yes the 3S-GE has an EGR valve mounted under the plenum chamber, close to the throttle body. It's bolted under underneath and has a diaphragm sticking out to the rear, towards the firewall. it looks kind of like a mushroom on it's side. But don't worry about the EGR valve, it really isn't important and only helps with emission control. Due to all the exhaust gases going into it it always becomes blocked after about 40,000 kilometres and no one ever bothers unblocking it as it makes no difference to the running of the engine. I bought some extractors for my car and they didn't have any where to connect the EGR pipe (runs from the EGR valve, along the side of the cylinder head and about the distributor, and connects to the exhaust manifold), so I just removed the EGR and junked it. The car runs fine. But if it makes you happy clean it out or buy a new one as it is sure to be blocked solid.
4) The engine does use a fuel evaporative system for excess fuel that is pumped through but not needed. You'll see a pipe running from the fuel rail to what looks like a large metal can bolted to the side of the engine bay. A second pipe will be running from the top of this can also. This can is the evaporative canister.

As for your engine problem. I would say that the problem is either with the cold-start valve or the cold-start injector. Contrary to what you will have heard it is possible, though very hard, to flood a fuel injection engine. I believe in this case you are getting way too much fuel in ratio to the amount of air coming in. That is why you have to wait a couple of hours before trying it again, to let the excess fuel evaporate.
The cold-start injector is for very cold conditions only. We are talking past minus and then some. It shoves in a great dollop of fuel when the engine is started from VERY cold. This should shut off almost straight away but if it doesn't and keeps going till the engine is hot it would explain the over fuelling problem. Also it may be kicking even though the car is not cold enough to warrant it. The injector is on it's own in the front of the plenum chamber about half way along. Before starting the engine, while it is completely cold, remove the wire coupling from the injector and the fuel feed pipe (tape the pipe up or you will get fuel everywhere when you try to start it). Try the engine, if it's starts fine then turn it off and reconnect the fuel feed pipe only. If it start ok again then the problem is the computer or sensor sending it information to activate when it shouldn't. If it doesn't then the injector is stuck open and the injector is the problem (when the engine is warm and running the ECU will be able to compensate for the excess fuel by lowering the amount to the other 4 injectors). Unless you live in the Arctic circle, Canada, or New Zealand (Baaa..aa), you can probably just disconnect the cold-start injector and forget about it, I am not lying the weather has to be real cold to warrant it.

The cold-start valve is a wax plug in the throttle body that is heated by the water system. While the engine is cold (the water is cold) the plug allows a richer mixture in, by regulating the air. If this was stuck on you would start fine when cold. If it is stuck off... well odd things happen.

Sorry I can't be of any more help.
I have attached a photo of the cold-start injector so you know where it is.