1. Does the new company need to do anything about my EAD?
No.
2. If the new company wouldn't transfer my H-1B, would it be OK for me?
Yes. Just remember to extend your EAD.
3. Again, if the new company wouldn't want to take the trouble transferring my H-1B, could I hire an attorney to do it?
If the company do not cooperate with lawyer, he can do nothing.