Bazs: Köszi! I wasn't aware of Hungarians in the forum!
Church:
First of all, I don't live in Budapest, but I know a few good places. In my opinion, you should avoid "fashionable" inns, and try some of the older, more modest bars. If you want to taste Hungarian beer, try
Soproni Aszok (Aces of Sopron),
Borsodi (My personal favourite

or
Borostyán (Ivy or Amber. I don't know the exact translation)
Places worth a try:
Wigwam, Zöld Pardon, Rádai Club, and the pubs/club of the Universities.
Oh! I don't know what kind of _harder_ drinks you tried in England, but if you don't know them, you should try:
Jaegermeister, Unicum Next (bitter drinks made of herbs)
Hubertus, Becherovka (also made of herbs, but tastes like Christmas :cool

Mézesbarack (A special -strong- hungarian liquor made of peach and honey.)
bigsamtheman:
Yes! The Hungarian Cup winner, and the NB1's runner up qualifies for the UEFA cup, and the champion can play to reach the CL group stage. (Usually they don't reach it...)
A little correction: French first league is also running. I forgot it.
Updates soon (i have 6 completed seasons) but sadly I can't write more because a great hangover...