1) If you want to exit, go to the "arrivals" terminal. The other one is the "transfer and departure" terminal.

Per your point (2), you could go transfers and go through transit security and wait a few hours, then exit. I have never done this, and there is no ready way to exit from the departures terminal, but others have reported that the agents let them walk backwards through security to get back to arrivals.

2) Not at all. There are basic chairs in the arrivals area, as well as in baggage claim, but I wouldn't recommend this. No real way to take a nap. With 14 hours, and a family, I would suggest a hotel. There are several moderately-priced ones between the airport and Souq Waqif.

3) They only take credit cards. Cost is QAR 100. There is a Qatar National Bank window on the right-hand side of the arrivals terminal. They may have some accommodation (like a pay card).

4) The only practical way to get there is by taxi. It should be around QAR 30-35 (USD 8-10) (flag drop from the airport is 25).