Speaking from personal experience mulling wine many times:
In general, you want a dry or semidry red, of innocuous flavor profile. You do not want anything with strong tannic, acidic, alcoholic, brett or sweet flavors; these will become concentrated while mulling and quite unpleasant. Your ideal mulling wine is an inexpensive, inoffensive, young merlot, burgundy, petite syrah, tempranillo, beaujolais, or other "middle-of-the-road" wine, maybe slightly on the sweet side. Cabernet sauvignon, chianti, rioja, and similar wines tend to be poor choices, although of course it depends on the individual wine. Also, look out for high-sulfite-added wine which also can develop off flavors.
I'll contradict Sarge here and say that you do not want a wine which is turning towards vinegar; you'll end up with a very sour crock pot full of mulled vinegar. However, mulling is an excellent thing to do with wines which have been oxidized (but not vinegared) and lost a lot of their flavor, either through being open too long or too long on the shelf. Certainly if you spend more than $9 a bottle in the USA for wine for mulling, you've made a mistake.
This is very similar to how you would choose a sangria wine. The main difference is that for sangria you want bright and acidic flavors, whereas for mulled wine you want heavier, darker flavors.
I've used both sugar and orange juice to sweeten mulled wine in the past, but obviously orange juice would change the flavour significantly. Add more sugar slowly, tasting as you go.
Best Answer
Yes. If you wouldn't drink it, don't cook with it.
Whatever it tastes like out of the bottle, it will add that to the dish. Cook with a wine you might pair with the dish (light wines for seafood, chicken; heavier wines for meats and stews).
Don't use a fruity wine unless you want your dish to have some fruit notes. Don't use a very dry wine if you're making a sweeter dish.