I use an HP T5720 that I got off of Ebay. I think i Paid right around $100.
yes, you can install remote management software. Most people use some version of VNC.
Check out this thread for more details on the thin client setup.
Help with thin client HP T5720 set up
I would recommend search for "thin client". There are many threads on what to look for and some tips.
Good luck!
Rob