Yes, it should. As you can see from the 'Terms' on the Facebook site, 'Facebook' is always capitalized.
The logo has nothing to do with if it should be capitalized or not. Facebook is a proper noun, therefore you capitalize it. It's odd that people would think otherwise, because you wouldn't put something in all caps just because the logo is in all caps, would you?
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You should press Ctrl+W to open up the Facebook menu. Remember this only works with Facebook open.
Go to your email (that you had signed up with) and there should be an email from Facebook with a link to confirm your account.
Facebook should not have offers popping up on the screen. If this is happening updating the application may be useful.
If you can't get on facebook the first thing to do is send a message to the facebook team, they should message back and tell you why you can't get on. You may not be old enough or friends who are not old enough. They should soon put your account back soon.
Putting your phone number on Facebook is not a good thing to do. If you want to give your phone number to someone, you should give it to them directly. On Facebook, strangers can see your information and use it unwisely.