just check!
they have recently changed the law and hotels, who have traditionally sponsored visitors, have to report people who are not staying there!
Better to get a visa in advance as there would be not be a problem.

Qatar has always been leary of single women under 35 years in case of visiting "immoral" people being allowed in! as if they couldn't be immoral at 36!

You should have no probs here as most people are very law abiding, as a man I feel safe taking thousands of dollars from an ATM machine. Women if dressed "provocatively" may attract unwelcome attention but generally it goes no further than that.