1. Yes, you can, provided you have maintained proper status until then. It is called H1B portability. ( read page 7 question 1)
2. Well, it can be tricky. You can transfer, if your first H1B with company A is still valid. Check with attorney on this.
3. It is a little complicated situation, I would check with attorney on this.
4. Not really as long as all the documents are filed properly.