I'm in doubt about the use of TBB, because I'm a month without using facebook for reasons of geolocation, I can utilize my TBB through the personal accounts because my registration in social networks are correctly registered in the country where I live with my address residential and mobile number. Vpn service is not a solution for me, because I have to register.
When you're logged in on Facebook, Google, ..., they also know who you are on other sites. Sites that have +1 or Like-buttons, sites that accept g+ or FB login for comment threads, etc..
From your question, it seems you don't want to hide your identity, just your IP/location. If so, this would not be an issue for you.
You can do that, but you loose all anonymity if you login. This doesn't only according to facebook, this is according all sites you have to login. You can only be anonymous if you don't login in any site.
of course. but maybe at first facebook or gmail prevent logging in without identifying yourself for example if you want to login to Facebook it'd ask you some question (i.e identify your friends by their pictures or sending a SMS to you) to protect the account from possible malicious users. actually I personally go facebook via Tor and it works fine