Not true, and Im living proof. my T mobile service WAS paid in full (bought myself out of the agreement) and it DID affect my credit score. even showed up when I wanted to rent an apartment. so did my bought of my Sprint agreement. You may be talking from a technical point of view, I'm taking from the what they do and get away with point of view.
No, you do not HAVE TO provide you SSN, However they WILL NOT provide you with any service other that a pay as you go phone without it. NONE. this again, was tried with AT&T (Who remembers their unlimited everything plan for 100 bucks?) , Sprint, and T-Mobile.
Edit: Oddly enough when I switch back to Sprint, they didnt refuse me service based on my canceling the agreement previously. this, most come to find out, is directly influenced by how moronic the sales clerk is, and your luck.