Am I right in thinking that although we will have new bodies and be like Jesus, there will be certain things we will not be able to do because He is God?
You are correct in assuming we will not become like God. But Paul said God wants Jesus to be the firstborn among many brothers (Romans 8:29). The firstborn has responsibilities, rights and privileges not afforded to his brothers, but aside from that is like them. And John said what we will be has not yet been made known, but when the Lord appears we will be like Him, for we shall see Him as He is (1 John 3:2). Beyond that, we’ll just have to wait and see.