Your Serial Key has been saved in your Home folder ( the folder with your user name, usually under C:\Users\YOURNAME ) when you originally activated the products. And of course, is also in your email.
The Serial number is the only thing you need to reactivate a product, since the download itself is always freely available here so, you can either insert it manually with the Addon Manager "Register Serial" function, or double-click on your saved .REG key, to insert it in the registry, which will save you from typing it manually.
Yes, of course you can install on multiple systems (as long as they are all yours), and there are 6 activations available, and if the last activation has been made more than 90 days earlier, additional ones after the 6th will be granted automatically.
Since you are building a new PC, and are interested in P3D, I strongly suggest to wait later today when we'll release all updated installers for all our products, supporting P3D V4, which should be released today too.