(1) If by public transport you mean buses they are utilized mostly by laborers. You can find taxis but it's difficult to do it on the street (relatively long waits in hot weather).
(2) Eating, movies, pubs.
(3) You can find some Tesco and Waitrose branded items plus the usual stuff in terms of cereals, canned goods, Mars milk drinks. It's not a literal desert here, Natalie.

You can probably get away with holding hands. You can give him a hug when you arrive and a kiss on the cheek is fine. Keep the affection for indoors. He's not going anywhere here.

Great that you've gotten a job. Doha's not too bad; it's not a great metropolis but if you're prepared to settle in then you will.

Good luck.