Yes, you should say that when entering a home. I never heard it used at work: knocking on the door is usually enough..
"Permesso" is also used when you are asking to someone to move out of your way (e.g. when walking in a crowded train) or when you need to pass through two people talking to each other (e.g. in a corridor)
Duo doesn't care about punctuation. So, "I can enter?" could conceivably be acceptable, just much less likely. That kind of construction is used sometimes, but usually has a little different nuance. "You are joining us?," etc. has a feeling of greater expectation that it will happen.