The dungeons are big, but its not forever.... and there are 3 levels to each dungeon (dont worry, level 3 is small).
and there are 3 dungeons in the first battlefield.
You will get quests from all the merchants. (as you mentioned 5). The solutions are not nescesarily in the first dungeon, it can be in any of the 3 dungeons... but most of the time the solutions are on the 3rd level of the dungeons (the bottom level).
If you have only found 1 battlefield key so far, you will only have 1 unlocked dungeon. When you find the other battlefield keys, they will unlock the 2nd and 3rd dungeon in act 1 (from act 2 its 6 dungeons).
hope this helps.