A few guesses:
- Memory not making contact (or damaged in 'transit') - push them down well.
- Socketed custom chips (my A3000 is 1500 miles away and can't remember which ones are) - try pushing them hard
- This hardware is old. There's a good chance that during the rehousing process some solder joint came loose. Look for contact problems throughout. Non-responsive mouse could be a short-circuit or a damaged CIA. Does the pointer move if you hold down one the amiga keys and the arrow keys?
- Remove all cards / other hardware and try to isolate the problem. You mention hanging upon boot off IDE. Have you tried SCSI? What happens if you boot off a floppy? Have you tried placing the Fast ATA on another zorro slot? Any incompatibilities between FASTA ATA and the A3000 (not sure it's a valid possibility, but worth a google search against your versions of Buster/Ramsey chips).
- What filesystem are you using in the HDD? SFS is known to place a greater strain on CPU and, in the presence of a flaky system overall, cause lockups. If booting from floppy works fine, try PFS on the HDD (freely available on aminet).
Good luck!