I can't seem to figure out how to grab objects in the 360 version, if you can at all? Stupid question, I know. Can anyone enlighten me?

Hold A when you put your reticule over an object or body, and then let go and when you're done dragging or carrying it, just press A to drop it.

Hold down the activate button

Ah! Thank you! I KNEW I'd feel really stupid after someone told me how to do this! Grabbing stuff was great fun in Oblivion, glad I can still mess around with it!

Stop trying to grab NPC boobs.

Lol!
It would be funny in an 8-year-old sort of way if you could grab some boob and stretch it halfway across Whiterun!

The only thing with the grab system in this game is that it tends to grab from the middle of the object instead of the edge you were aiming for. This can be troublesome when you're delicately trying to set up displays.

Or trying to place Baskets and Buckets on NPC's heads....

Admittedly, I haven't yet done that, but I plan to.

It's just annoying when I'm trying to set a weapon leaning against a shelf or something, and I have to smash it off of other object just to get it angled the way I want it to.

Also on a semi-related note, Dwemer bowls/plates and whatnot don't act the same as regular ones. They have no sides, apparently as things just fall right through them.