The only time I saw a legal way of using a Windows 7 key and license on more than 1 PC was the special offer of Retail Family Upgrade Pack to Win 7 Home Premium for 3 PC's. ( I have not seen them advertised recently though some can be found on Amazon etc for $ USD 256)
Or can it be installed and re-installed infinitely on the same machine?
Yes it can be reinstalled indefinitely on the same machine. You also should be able to change minor parts in that PC without having any issues. But change the Motherboard, and it will require a new license if using OEM.
With Retail(the expensive version) you can change all of the parts and activate it with no problems.
Also, how many machines are you able to simultaneously have W7 installed on? Is it just the one?
Just 1. The family pack might be the only exception. I say might, because it could be 3 keys, I haven't seen it so I'm not sure. It also is no longer available.
1) Minor correction. OEM version an Be re-used on a New MB if the orginal MB dies, Can not be eused if Orginal MB is just being upgraded to improve performance or add features. This goes away with Windows 8, Currently Win 8 OEM removes the restriction of being tied to first MB installed on.
2) Number of Re-installs on same system is generally unlimited, however to many over a specified time will trigger a "have to Call" or If you have the 3-system family pac. Only ever had to call once (it was a re-install of family pac) - Was NO problem they only asked if it was installed on more than 3 computers.