In theory, the C# in C is a minor second (it is part of the locrain mode, for example), a very weird sounding note. So according to theory, you might play it (even in myxolydian) if the vamp you're playing against has dissonant chords or a flavor which calls for that kind of notes. But you won't have the gig for Britney SPEARS tour...
For the second question, the second thing you wrote is the correct one, if you play with a piano for example( who can't be detuned) if you play with a bassist, just tune him down and play as you would normally, you'll have a deeper sound.