What would a mug of ale or a roast chicken cost in a medieval tavern/inn? How accurate is fantasy when it represents taverns? Are there examples of things like menus or prices and how would they match up to westeros?

The main inaccuracy in fantasy is A. how many of them serve food or have rooms to let and B. how many of them there are compared to places that just serve liquor. 

Historically speaking, because there was much less travel in the (especially in the early) Middle Ages than in later periods, there were far fewer establishments that catered to travelers, unless you were on a major trade route or pilgrimage or the like. So what you had was a lot of taverns that just sold alcohol and nothing else, because people were expected to eat at home, and people were expected to sleep at home, and pilgrims were expected to find lodging in monasteries. 

As we get to the High and Late Middle Ages, more and more people were traveling for business and other secular purposes, and the number of pilgrims had outstripped the supply of beds in monasteries. There was more demand for more services, so you started to see inns (as opposed to taverns) popping up that provided rooms to let, and those places tended to serve food to travelers.  

So to focus your question on prices for booze…in the 14th century, ale ran about 1 to 2 pence a gallon. Converting that to dragons works out to about 1.3 silver stags to the gallon in Westeros.