We booked our flights to Cancun and decided to stay a week in Cozumel and week in Playa del Carmen.
I think that is an excellent plan. Some Cozumel fanatics aren't interested in anything else, but there is much to enjoy on the mainland peninsula, too. Do ruins interest you?
Mayan Ruins Mexico: Yucatan Peninsula Mayan Ruins, Mexico - LocoGringo
1) What diving base and hotel do you recommend on Cozumel?
There are many good choices, but it really depends on what's important to you. I'd suggest that you shop the hotels on Kayak.com or Hotels.com, then research the reviews on your favorites - here, on Facebook, on Tripadvisor.
2) Would you rent a car in Cancun and use it everywhere or would you use a scooter in Cozumel and the bus everywhere else?
You may well want to rent a car in Cancun from one of the agencies that offer free dropoff in Playa del Carmen for that week, but that would depend on where you chose to stay and where all you want to go on the mainland. You do want to take full insurance from the agency; don't count on your credit card or other coverage.
I would not take a rental on the car ferry; leave it on the mainland. Riding scooters on Cozumel is one of the most dangerous activities possible, so please don't. If you stay in town, there is little need for a rental car most days, but if you stay at a southern resort you might find it helpful. Fiesta and Isis seem to be the most trusted on the island.
3) Does it make sense to go for AI in Cancun or are the restaurants there very nice and we should consider only breakfast at the hotel?
There is a big difference in Cancun and Playa del Carmen. For the latter, I'd go for breakfast in, then eating out. AIs are beneficial mostly to those too lazy to leave the resort or with kids.
4) Is Nitrox widely available in Cozumel?
Yes, usually $10 USD per tank. For your hotels and dive operators, you'll just have to ask if they'll accept payment with German currency at a reasonable exchange or credit card. For dining out and other expenses, it's best to use Mexican Pesos obtained at a
bank location ATM (beware the merchant and street ATMs) with a
debit card withdrawal. Also see this thread:
More photos of the bank ATM scam
5) Is the Palancar reef the "best" diving site in Cozumel?
There are many fans of many different sites. We have a developing situation tho with news to be announced later today. We've been discussing it in this thread:
Possible southern reef closure
6) What diving base do you recommend in Playa del Carmen / Tulum for cenotes?
7) Is it really possible to see bull sharks near PDC in march?
Sorry, I wouldn't know. You may want to ask on the Mexico forum.