When replacing the hard drive in my Dell Latitude E5400, what it the largest capacity drive the unit will take?

If the RAM slot you used was empty before, it could well have oxidised contacts and be full of dust etc. I would reseat all RAM twice before reassembling. If this doesn't fix it, I would suspect faulty RAM. If you suspect overheating, you'll need something to excercise the CPU. I've had one of these fail when applying OS updates purely because the fault only came out when it was working hard.

The CD/DVD Drives in these machines seem to be prone to failure, so the unit will need replacing. Out of 6 machines I recently came across, 3 had failed optical drives.