You can stay at hotel serviced apartment's for less than $5-6k.
(I stayed this year at Radisson 1BR premium suite for 10k AED/month but at a distant location)
Just look at hotels for long term stay option offers (or ask them if they can provide).
I also stayed a month for (560 AED/day) or about $4,5k in a massive 90sqm2 serviced apartment in Dubai Marina.
Depends on what location are you after and what comfort are you looking to get.
If you plan on driving:
Rent-a-car: ~2,000 AED gets you cars small SUV like Duster, Creata or Kicks or ~3,300 AED you can get Outlander, Sportage, Xtrail, Koleos
Petrol: ~200-400 AED (depends on usage)
Salik: ~100 AED (road tolls, depends on where you are and how often you drive on SZR)
You also need to include parking (some public ones are paid using SMS and costs 3AED/hour, crappy garages in buildings are 5-10 AED/hour, better parking at Blue Water islands, JBR etc are 20 AED hour or you get 3 free hours if you spend 100 AED in restaurants there just give them a ticket to validate it when paying the bill)
If you have emirates ID you (you are a resident) you need to swap your driver's license.
It usually costs ~1,000 AED (details
New to the UAE? Here is how to convert your driving licence to a UAE driving licence).
It's one time fee and the first drivers license lasts 2 years and after that one the renewed one lasts 5 years (and is super easy to renew at the ATM-like machine, literally takes 2 minutes).