1) Best mobile carrier; Qtel or Virgin? (include the plan your on please)
Consider making calls while overnighting in other countries and while in Qatar itself. Also international rates. Any hidden traps to them?
Qtel is the best regarding coverage in Doha. I can also make calls with it from around the world. I have pay as you go and is very simple.
I already have my own phone to use, which will require internet access (smart phone)
Ask about that in a Qtel shop when you get here
2) Best TV and broadband plan?
Depends where you live. I have 8GB through Qtel. Best to enquire when you get here
3) Best Bank? And what account seems to be popular for expats?
Consider including credit cards and purchases from overseas as well.
IBQ and I use my credit card to pay for hotel bills down route. I send money back to my home using the on line internet account, could not be simpler