When a cruise ship need a tender ship to transport passengers, does cruise company pay for both the tender and docking fee?

You should paid that in advance  when you booked on your cruise and decided if you're going to go off the ship and go on touring or sight seeing something like that.

On some ships, the tender boats are the same as the lifeboats used in emergencies.

If they anchor off and ferry passengers ashore, they either use their own boats or hire local tenders. Depends on how many wish to go ashore, and the port in question.  They won't be paying docking fees unless they dock, but will doubtless have to pay pilotage and maybe a fee per person landed.  Here they have to pay pilotage for the pilot to put them in the right position to anchor within the port limits and they usually hire local boats as tenders.

No docking fee if no dock is used.

Yes, of course. They contract with local companies- but the passengers don't pay to be transported on the tender. The cruiseline pays out of the money the passengers have paid to be on the ship. 

Most have long term contracts with the providers.

they hire the tender ..the tenders fees will include any docking fees ...many years ago i was a deckhand on a tug tender