I've got a peacock eel and a spiney eel. Both will get to be around 8 inches. They shouldn't be in anything smaller then a 20 gallon. They are not aggressive fish so they should not be put with aggressive fish.
A betta is fine, they will not bother one another. I had a betta in with my two eels with no problems.
I would not treat your tank with ich medication. Eels do not have scales and are very sensitive to the medication. If you have to treat for ich, use salt. Here is a post on using salt to treat ich.
http://www.aquariacentral.com/forums/showthread.php?t=39759
My eels won't eat shrimp, alive or dead. But every eel is different. In fact the only thing mine will eat is black worms that I get at the LFS. Mine love them. Also, what kind of substrate do you have. You should make sure that it is not rough and that it doesnt have sharp edges. This is becuase eels like to burry themselves. I would invest in a decorative half log,cave, or PVC piping of some kind. Eels love to hide and this would give yours a place to escape to if the other fish in the tank start picking on him.
You definantly need to cover any, and I mean any, opening on your tank. Eels are escape artists and they will find away out of the tank.
Eels can be such fun fish. Some people train their eels to eat right out of thier hand and others pet thier eels! One of mine is very shy, hiding all day, while the other is much more active swimming about and exploring. Feel free to PM me if you have any questions about eels.