you need a mod. example: with the more creatures mod you can pick up rats, mice, and tamed cats.
Destroy it, then pick it up.
Unfortunately you can't pick up cake when it has been placed.
Use a bucket
Mine it using a pickaxe (wood or better), then pick up the drop.
If you are looking for a way to pick them up, well, you can't pick mob spawners up. You can only destroy them with a pick but it can give you a lot of exp.
Destroy it, then pick it up.
Unfortunately you can't pick up cake when it has been placed.
Use a bucket
Mine it using a pickaxe (wood or better), then pick up the drop.
If you are looking for a way to pick them up, well, you can't pick mob spawners up. You can only destroy them with a pick but it can give you a lot of exp.
Craft a bucket and Right-click on the water with the bucket.
No, they only pick up blocks that are in the world
then your minecraft is bugged...
You will need the 0.3.3 update.
So far there is no trading system in Minecraft so all you do is just throw the items on the ground and they can them pick them up.
They can pick up endangered animals or plants, and I suppose if a rare jewel was lying around, they could pick it up.
Minecraft is a sandbox game. You can pick up and manipulate almost every block, travel in any direction for as long as you want, and do whatever is in your capabilities.