I'm in the imp village now, and I find that I cannot go the the Battlefields anymore!
normally, the button to enter the BF can be found at the secondary skill, but when I go there, the button is gone?

Well, is that because you're not supposed to go the BF in act2 (strange)
Or could it be that I loaded the three BF keys I found so far on one of my summoning dolls?

I want to go to the Battlefields!!

Each act has different battlefields.
To enter the battlefields in act2, you first have to find an act 2 battlefield key. And by the way, you don't have to take the battlefield keys with you. Just clicking on them once is enough.

I have the act 2 bf key. At least... I thought. I have three keys. Or do you have three BF keys for act 1, and need another three for act 2?

That's right.
Well, actually you have six BF keys for act 2, but you get the picture...
The different keys are for unlocking the different dungeons in the BF, not for opening the BF's in separate acts.

With a recent patch installed, the act 1 BF keys should have been removed from your inventory when you switched acts. The first act 2 BF key is in the prison on the citadel island, and the second can be found in the imp village. If you found these, try clicking on them, if you had just dragged and dropped them into your inventory before (you don't need to keep them, BTW).

I've gotten 4 out of 6 dungeons devoid of bosses. This random business is pretty stupid. Also, in one of the dungeons all I found on the third level was a 'parchment'. What is this for?

The parchment was the object needed to solve one of the quests.