That sounds like a lot of fun.
I used to have an AE102 that I put a 3SGE in (in Toronto).
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As for axles, I used ST162 axles, they fit right in. I should add that mine was a Gen 3 motor and tranny IIRC. I didn't have proper mounts, so I used a combination of mounts that I had kicking around. I have heard that a Caldina ST215 mount can be used at the timing belt end, but that is pure heresay, so take it with a grain of salt and don't spend big money importing mounts. I can get a picture if that seems like a help.
I actually used the axles to center the engine left to right when welding up the engine mounts. (by comparing the relative positions of the outer joints with the axles fully pressed toward the engine (inner joints fully plunged inward).
I put 48,000 kms on it like this, so I don't think you will have an issue. I did originally consider widening the track a little at the lower control arms, as the track on a Celica was marginally wider (About 1 cm IIRC). Also, I kept stock height, so if you plan on lowering it, widening the track to accommodate the axles may become more important. Funny enough, it sat at basically stock height with the stock Corolla springs.
Is LSD important? That is up to you. Mine would have definitely benefited from it, but there was always something better to spend the money on. Finish the swap before spending the extra on an LSD, you may well find that the swap is more labor intensive (and costly) than you anticipated. Are you good with a MIG welder?
I had a few minor firewall clearance issues. That was mostly due to where I put the engine, I tried to leave a lot of room up front for a turbo. A/C was all but an impossibility with the intake manifold that my motor had (because of where the AC pipes come out of the firewall on the Canadian spec (LHD) vehicle).
I used an ST165 rad because it could be fit into place fairly easily, and the price (for a new one from the dealer) was reasonable ($300 Cdn). The hoses came off in the right places, and it had the nipple on it for my planned turbo (which I never got around to).
Also (for the Auzzies) Canadian AE102 are little different than an AE101. Over here the distinction seems to be the engine displacement. They are not superstrut equipped.
Hands down this is the swap to do over a 4AGE 20V. I have driven both, each for 2 years plus, and the 3SGE wins in every metric.