The only uncertainty would be around the Omani visa. Supposedly you can enter Oman without paying for the visa if you arrive from Dubai. I would ask at the airport on arrival. Worst case, you have to pay 5 riyals.
Unless the process has changed recently, you queue at the currency exchange to pay for the visa, receive a receipt and then go to an immigration officer. If you already have riyals, you can pay at a machine and receive a receipt.
You can enter and exit the UAE as many times as you like and there are no border controls between the individual emirates. You'll receive a free 30-day visa on arrival which will cancelled each time you leave.
For Abu Dhabi only, there is an immigration form to be completed but it seems to be purely for statistical purposes.