1. Can I find a replacement shell of some kind to hold the motherboard from an old HP Probook?
On eBay, if you're lucky. It might be better to do something DIY.
2. What would be a good SSD for under $150 and at least 120GB?
Check to even make sure the SATA interface is good enough for an SSD. If it's SATA I, you're not going to get much out of an SSD. SATA II, don't spend too much money.
3. Would it harm the the parts to run it with no shell, if #1 turns out to be impossible?