Hmm, your shifter jumping out of first gear could be your clutch evidently. I recently did a lot of reading about clutches while I was shopping for a good clutch (settled on a stage 2). One of the articles I'd read was about bad clutches popping out of gear under load. But it is likely your first gear Synchronizer has gone bad and is causing the shifter to jump out of place. That would likely be accompanied by a grinding noise and a hard jolt from the car when you try to put it in gear.
At any rate, I have read that the Celica GTS 6 speed transaxle (which is what you have if it's all stock... transmissions are different) will bolt directly to the 1zz-fe block. However, you won't get as much power at the wheels since that transaxle is designed for an engine with different bore and stroke measurements. Toyota's parts are pretty interchangeable. I just dropped another 1zz-fe engine into my 2000 celica after her original 1zz-fe died from piston ring failure, but the 1zz-fe engine I replaced it with came from an 04 Corolla. Incidently, in 2012 I swapped my transaxle as well after my 3rd and 4th gear synchronizer went bad and that transaxle came from a Corolla as well. So, it doesn't really matter which car model it comes out of. It could be a Matrix, a Corolla, an MR2 etc.. what matters is, will this transaxle bolt onto your 1zz-fe engine?
A solid online resource is Monkey Wrench racing.
https://www.monkeywrenchracing.com/product-category/celica/celica2zzswap/
