Big mistake. When you cook with alcohol, the ethanol evaporates and the flavors concentrate. Hence, if you cook with shitty alcohol, you're gonna get a load of concentrated shitty flavors in your dish.
Never cook with any alcohol you wouldn't drink. This goes double for wine.
The ethanol evaporates, but only marginally faster than water would. Unless you're reducing it to a sauce, the quality of your wine (or beer, in this case) will have little to no effect on your cooking. Rather, the qualities you're actually looking for from cooking alcohol are the ability to extract the alcohol-soluble flavors from what you're cooking and possibly changing the pH of your liquid. Besides, when you're talking about a crappy American beer, 75% of the problem is the weakness of the flavor, not the flavor itself. Whatever concentration of the flavor there is would actually help the darn thing during the cooking process
ADDENDUM:
Interesting fact- did you know Shakespeare invented the word Assassin? It comes from the word hash, because Arabic assassins used to smoke a bunch of reefer before they killed someone.
Once the conventional wisdom, but lately refuted, seeing as there's actually no evidence that the Hashshashins ever used marijuana beyond the supposed etymology of the word.
Which, seriously, can you imagine getting ridiculously high and then deftly sneaking up to someone and murdering the fuck out of them? Has this ever happened before?