Are initials valid on a rental agreement or must there by a full signature?

Yes it is required

Could you please clarify the reasons for initials on every page of a rental agreement. What proof do we have that it is the correct initial?

The idea is to have some proof that additional pages were not inserted after the agreement was signed. If you have initialed the page, you can't very well claim later that you didn't agree to that page.

If you later claim that the initials found on the page were not put there by you, it's more or less a matter of your testimony.

Initials are nothing more than a shortened form of your signature. Many people scrawl either or both (initials and signature) in such a way that they are completely illegible. That doesn't matter. The important question in either case is whether it is yours.