Not usually, no. In the past, arborists used tar, but nowadays they just leave the cut exposed. That's what we do at home, and haven't had any troubles.

Yes. I use this paste (the stuff at the top of the page)

I'm not sure if you need to cover it or not, but be careful you don't cut the limb flush to the trunk; let there be a little nub.

Thanks! I guess fungus and insects are about the only concerns. Bansai enthusiasts seem split on whether to seal or not.

What mrmoonpie and Specklet said. It is also the way it is taught in the Master Gardener program. Here is one link on using the 3-cut method.

In the master gardener program I took the mantra was that trees seal not heal. Covering the wound makes it harder for the tree to seal the "wound". A flush cut past the branch collar will more than likely harm the tree.

Another link that mentions how to prune and avoiding sealing paint. This quote is from that page:

Pruning cuts do not need wound paint or sealer. Trees seal themselves by setting up a protective boundary between injured and healthy tissue in a process known as compartmentalization of decay. A good pruning cut will seal in a doughnut shape as the branch collar closes around the wound.

I am so glad I read this, I've done some work on my eucalyptus and was worried about THIS EXACT THING!
Bunny knowledge is awesome.