Do I always need to peel the avocado, or can I just chop it?
Some varieties of avocado do have edible skin (e.g., "topa topa"); however, such varieties are rarely cultivated since they are extremely perishable. The vast majority of avocados that are marketed in the USA have skins that are too thick for consumption. If you are in doubt, just cut off a small piece of the skin and chew it. If it feels like you're chewing leather, chances are you'll have to peel it. Eating the skin is harmless unless you're a domestic animal.
Sure you can. My favorite brunch is cutting an avocado in 1/2, scoop out a little extra then add an egg in each 1/2 and bake till desired done-ness which is about 20 min for a done yolk. This softens the skin of the Avocado and makes it quite pleasant to eat. Add your own twist but I suggest you top with cheese and serve with a hash brown or grits. Nom, nom, nom!