For starters that screw you're adjusting should not be all the way in. That's your bypass air and you'll want to insure that it's a couple turns open... depends on a few things and I don't recall any of the facts to repeat them... but with everything hot, set the idle to where you're wanting it to idle with that screw. It should be a couple turns open... it's actually supposed to be a measurement, like having it open a certain 'gap' - the computer is trying to take care of everything else and in essence it really isn't intended to be an idle speed adjuster. Messing with it just tends to muddle your driveability of the thing.
Things like A/C when it's engaged bump up the idle. Any air leaks bypassing the throttle plate will increase your idle. Those two hoses running from your power steering pump to the intake, bypass air to increase rpm when required. I personally don't see the point and it's only asking for an old hose to leak...
If you need a tune-up, new plugs whatever - do it. But if all is well, I wouldn't be pointing blame at hard parts like idle air control valves or whatever. Check hoses, check wires. I had a broken wire on mine one time that I didn't find until after I'd been over it visually several times... it was to a tempature sensor and it was all that was wrong even though the problem had me convinced that something more serious had failed.
Good luck, if we wind up discussing it further maybe some more specifics will occur to me to mention...
Kevin
btw: The '85 Celica that I have had didn't hardly idle high when cold at all... but I never really bothered with it. I think the early 22re had an electromagnetic idle air control valve that can be removed and cleaned... the later ones had purely a mechanical one not unlike a thermostat and there isn't much chance it would need servicing...