I noticed none of my display cases build in my new 3 manors doesn't work. I can open and close it but I can't "activate weapon rack". The only way I can put weapons in is by dragging it with "E".

Does anybody else has the same problems?

As far as I know that is how it is supposed to be.

The "classic" display cases in Skyrim could only display weapons (by putting your equipped
weapon into it). The display cases in Hearthfire are for displaying anything that fits into them,
like gems, artifacts, rare books etc.

It's a little tricky to put and arrange items in them, expecially since the items tend to leave
the cases (I reported that bug right below your post

You really think so? That its intended that way?

I am not 100% shure, but this is the only answer that makes
sense to me.

In the trailer for Hearthfire you can see several display cases with
several stuff in them (troll skulls etc.), so I think these cases are
not intended to hold only weapons like the "normal cases", and
therefor don't have the "activate display case" option. Other
cases in the game, like in the archmage-chambers or in the
dawnstar-museum, that don't hold weapons but books, soul
gems, shields etc. also lack the "activate" option.

I had prefered the classic cases over this one, or at least a better
way to place items into them than dragging them there. Expecially
since the items don't stay in there and tend to fly across the room
as soon as you reenter the corresponding building.

You could always manually put in the cases whatever you wanted, the activation option for the weapons was just to make it easier to use them.

I agree, I think it is by design just like a PP mentioned, it is like this in the ArchMage tower. But if they made it something akin to the http://skyrim.nexusmods.com/mods/10891 Mod but for Display Cases would have been nice. Same goes for regular shelves, tables, etc.The whole standing back and throwing items into the hole is very frustrating to say the least. Using the Activate button and moving items after you have dropped them out of your inventory has never worked because once the room resets items fly across the room like a possessed rabbit trying to kill you.

I've only encountered a few display cases that have the 'activate' option. The vast majority require you to drag and drop stuff into them. I just assumed this was how it was supposed to be - to allow you to put miscellaneous items in them.