It depends, and if ever it is possible the family member that you would transfer your mortgage to, would be liable for the repayment of the debt of your mortgage.

The property is subject to the mortgage. In order to sell a portion free and clear of the mortgage you would need to obtain a partial release from the bank. The partial release must describe the portion you want to sell and must be recorded in the land records.

Yes. If the bank has a mortgage on the property there is a due on tranfer clause in the mortgage that the property owner signed That means the bank must be notified of any transfer of ownership and it can demand payment in full of the mortgage if any transfer is made. A quitclaim deed would be a transfer of ownership.

You can't 'transfer' your mortgage to another property. The bank owns the mortgage lien. You would need to negotiate with the bank to modify its lien.

You cannot transfer a mortgage since the mortgage is owned by the bank. The bank is unlikely to remove the obligation from you to an LLC. You would need to pay off the existing mortgage, transfer the property to the LLC, and then refinance under the LLC . . . if the bank will allow the transfer of title and new mortgage.