Emulsifier: is it safe to cook mayonnaise?

The mayonnaise itself is only a stable emulsion in certain temperature ranges. If you are talking about standard mayonnaise (made with egg yolk), it will split when heated, and won't help your new sauce at all.

If you are planning to use one of those vinegar-oil emulsions sold in supermarkets under the name mayonnaise, I cannot predict how they will handle. They may or may not split. But they are not in principle supposed to be heated, so you can experience weird results. And the emulsifiers in them are calculated to be in the proper ratio for the mayonnaise as sold, so there will liekly be too little of them to sufficiently emulsify a sauce when added to it.

Mustard won't help you at all. It is only a physical emulsifier, so only works with limited success. The cayenne flesh is already doing some physical emuslification, if that's not enough, mustard won't make it better.

All in all, if you plan to heat your mixture (why?) then mayonnaise is unlikely to help you as an emulsifier.

Mayonnaise is safe to cook. Prepared mayonnaise that you purchase in the store has been pasteurized and usually has more than enough acidity to keep any harmful pathogens from growing. Cooking with it can only make it even safer (if that is possible). If you need more evidence of how safe mayo is, you can read this.

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).

Safety isn't the issue here. If it's safe to eat cold, it's safe to eat hot.

That's my answer to the question posed. If seeking advice on how to emulsify your hotwings sauce, cook it as normal, then after it is taken off the heat and cooled a little, whip a little mayonnaise into it.

Flavor wise I think this won't affect the hot sauce because the main thing you taste in mayonnaise is vinegar, which is probably already present in your sauce and if it's not then the vinegar would essentially behave as a seasoning, not a major flavor profile ingredient.