>want to know is, can I swap out the LM350 for the LM338? If so, >would I need a different transformer?
Yes.
No. The LM338, if you check the datasheet, works with a wide voltage range.
>Can I mount the transformer in the case? Any EMI or grounding >issues?
Probably some EMI. But you can use a metal casing (carefully isolated from any live). They're pretty cheap, and look good

, most electronic shops have them . Or use a
toroidal transformer. They generate less EMI. But again, expensive.
>Could I use this device to turn on both the pump and the
>transformer? Pump Relay Switch Kit
I think so. But I dont like that relay switch kit. I think it has several security issues, specially with unprotected live leads.
However, you can just buy a 110v / 12v relay capable of supporting up to 10A. Stick a fuse on it and "tada". Connect the 12v to the PSU and... auto start. It's what i did. Just powers my Eheim 1048 anyway.