If I purchase this downloadable software will I be able to re-download it, if lets say I buy a new computer?
If you purchase the program as a standalone, yes. If it's a part of Object Desktop, you'll only be able to download while your subscription is active. If it's expired, you can still install and activate as long as you have the installer, but you lose the ability to download.
If so, how? Also how many computers can I have this program on?
The EULA allows for up to two, as long as you're the primary user.