To answer your question in bold: absolutely. ARD, Timbuktu, and VNC all do basically the same thing at the core: let you see and manipulate a remote computer's screen. Yes, remote keyboard and mouse control work.
Apple Remote Desktop is really more of a workgroup management software. It does indeed let you take over a screen, but its strong suit is letting you monitor a whole computer lab's screens at once. (ARD also has other lab management functions, like letting you install software on all the machines at once.) ARD really looks like it was designed with the classroom in mind.
Timbuktu is not designed for lab monitoring, but it excels at screen control. For example, drag and drop between screens works. Also, Timbuktu works better for non-LAN usage. Its performance is acceptable even over relatively slow Internet links.
VNC works, but it's slow even on a 100Mbit LAN. (VNC on a LAN is about as fast as Timbuktu over dialup!!!) VNC also lacks things like drag and drop and other niceties. But it's free.
I use Timbuktu over the internet to help my relatives (who all live 600-3000 miles away) with their Macs.
As for running headless: you may run into the problem that the Mac won't boot with no monitor attached. If you do encounter this, spend the $20 for a
Dr. Bott gHEAD device, which mimics a monitor, allowing the Mac to boot with no display.
tooki