Walls – What drill bit do I need to screw a hex cap screw into a wall stud? The magnetic DeWalt nut driver bits aren’t an exact match

Yes. It's OK

Mounting holes do not significantly affect the strength/function of studs (within reason - if you make Swiss cheese out of them, you might manage to have an effect - but that's not what you are describing.) If you are going to be very close (1/4" or less) to the old holes, you may need to drill them out and glue in a dowel before you drill the new holes - if further away, no need.

you definitely don't want to drill into the brick. Drilling into the mortar isn't too bad though, and easily repairable if ever desired.

I've used Tapcons for anchoring into mortar. They work well:

The box will say what size masonry drill bit you need to pre-drill, then you just screw them in with your driver like any screw. I'd suggest longer is better (though no longer than the width of the brick, as that may be as thick as the brick is.)

Also, make sure this brick you are drilling into isn't a chimney. If it is, make sure it's either not in use, or has a metal flu inside (and don't puncture the flu)

Your idea of installing a wood cross-member into the brick and then hang something from there isn't a bad idea. I'd suggest 3 screws, then perhaps attach peg board to that, hanging the pots from peg board ala Julia Child