I can tell the difference at 12. There's a flash test at http://jakemandell.com/adaptivepitch/ and the (probably heavily sample-biased) histogram at the end suggests 10-20% of people can't differentiate 1/4 of a semitone and 60% ish can't differentiate 1/12.

Yup. Have you tried a higher octave? Depending on the timbre, usually a higher octave will be more sensitive. At least for me.

Yes, easily, (even) between consecutive notes. Try playing two notes, then the same ones in reverse.