just two questions

could anyone anwer this questions:

1. suppose I'm playing the C mixolidian scale and I want to bend any C note, can I bend it half step? cause if I bend it half step it would sound like a C# and that note is not on the scale. (please don't tell me things like "if it sounds good for you do it" tell me if in theory is right or wrong)

2. If I have the guitar tuned half step down, how do I play the C mixolidian scale in the e string:

like this: e__8_10_12

or like this: e__9_11_13

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.