Regarding the rentals, in Dubai they have short and long term rental.
Short term is everything below 1 year (on day or 9 months - doesn't matter) and with these you don't pay any bills like electricity as it's all included in the rental price. Even if you make an agreement with the host (hotel or apartment) that will not be considered as a valid address proof in many places and banks. They will ask you for either Ejari or DEWA/utility bill.
Long term rentals are 1 year or above. You will sign a
tenancy agreement and you will have to register it with the government (
Ejari). Once you have Ejari you can have electricity, water, AC in your name (
DEWA). Also note that land lords would ask you to pay 6 or even 12months in advance (1-2 checks) and would rarely go with 3-4 or 6 checks/payments and it would definitely raise the price as you can cancel the agreement, pay one or two month's rent as a penalty and you're out. The DEWA is important as it's a utility bill that you would be asked as an address proof. To sign this contract you need to have EID and UAE bank account/check book as it's very common that the payments are made by checks (not by bank transfer).
Also, owning a property there doesn't mean you live there as if it's rented the utility bill will be in the name of the person that rents it so you won't have the address proof again...
You won't be needing the UAE address for the whole process of applying for the residence... For personal bank account opening, most of the banks will just need EID. Address is not that important for them. For Business bank account they might ask you to have either rented office (no flexi desks) or rented flat so you have at least one address where you would pay utility bills... Or they might ask for both... depends...
You mentioned that you need address to get the post. If you just need it for that you can rent a PO Box. As it's so hot there you won't see a mailman going from building to building delivering mail. In most cases they have buildings with dozen of PO boxes under AC and the mail will be delivered there. Most of the things will be delivered to you either electronically or either by courier that will ask you for your current location and will deliver you the parcel there (regardless of what's the address you have put).
I'm writing this from my personal experience... Hope it helps...
If you stay in a hotel or apartment you can get your mail there WHILE you are there... Once you move out your mail might just bounce...