Yes, you can safely cook mayonnaise, but it will most likely separate if you apply much heat. Therefore, it is unlikely to help much with thickening or emulsification in a hot dish. While you will find mayo in some hot dishes, it is usually applied as part of a mixed sauce after cooking is complete. You could certainly base a wing sauce/dressing on mayonnaise, but you wouldn't want to cook the sauce.
Traditional wing sauce is made up of equal parts melted or clarified butter and hot sauce, though you can vary the ratio quite a bit. Cayenne pepper can be added to increase the heat without altering flavor much. Frank's RedHot is the original in "Buffalo wing" sauce, but pretty much any hot sauce will work. Sometimes palm oil or a hydrogenated oil is substituted for butter in commercial sauces, particularly shelf-stable bottled brands. If you wish to add a thickener or stabilizer to your sauce, I'd suggest xanthan gum, which is readily available, relatively easy to use, and is used in many commercial wing sauces.
I don't think mustard would be out of place in wing sauce at all. In fact, I've had some good sauces with a heavy mustard component. It's more for flavor than emulsification, though.
My own preference is to whisk together equal parts melted butter and Frank's RedHot, Crystal, or sometimes Sriracha. I've never had a problem with it separating. I usually give my wings a second toss in a homemade Cajun seasoning blend of garlic powder, onion powder, dried thyme, dried oregano, paprika (preferably smoked), ground celery seed (optional), white pepper (optional), and lots of fresh ground black pepper (not optional).