Elements on a second computer

If by activation code you mean your serial number you should have printed that page and put it in a safe place, as well as saving the information on your computer... if you purchased via download you should also copy your install file to an external device in case your hard drive ever dies

You then install and enter your serial number exactly the same way you did the first time

Yes you need the code you find it in your email or in your Adobe account the same place you found it the first time