Hey Alan,
Airport Transportation:
http://www.iaa.gov.il/Rashat/en-US/Airpo…
The train runs from 3am until midnight and is perhaps your best option (it should cost about $3-4 per train ticket). Once you arrive in Tel Aviv, you can take a cab from the train station to your hotel. Cabs run all night and are easy to find and it shouldn't cost you more than $7 to get from your train station to your hotel, just be sure that the cabbie turns on the meter.
My own personal preference for transport is train b/c it offers good, inexpensive, reliable and safe service. Busses are worth considering to as a method of transport from aiport to city, but they might be more expensive then the train and more confusing to use.
There are transportation options at all hours, even on the Sabbath, so don't worry.
It's pretty much impossible to upset the locals. They are very open-minded, and if anything, might upset you with their lack of political correctness and their tendency to 'talk' to each other in loud voices that border on yelling. Although, I would avoid talking about politics unless the subject is brought up to you. If you're views tend to be more pro-Palestinian, it could make things a bit uncomfortable and awkward when discussing politics (not that people will mistreat you or anything for expressing your views).
There are tons of good bars/restaurants in Tel Aviv! I don't know too many as I don't eat out so much, but among the recommended places to go: Spaghettim (excellent Italian food), Osteria da Fiorella (excellent Italian food, but go for dinner), Orca (great seafood), Okinawa (great, reasonably priced Japanese food), Max Brenner (chain of famous chocolate restaurants- go for their incredible selection of deserts), Gelateria Iceberg or Gelateria Aldo or Gelateria Sicilian all have ice cream that is as good as any you'll find anywhere in the world, Thai House (excellent, authentic Thai food), Cafe Neto (a chain of excellent coffee shops that serves very good meals and wonderful deserts). Many good restaurants are located in the Marina (in Herziliyya Pituach, north of T-A, and in Neve Tzedek neighborhood inside T-A). Food in general is reasonably priced. While cheaper than food in Western Europe, it's not very cheap (expect to pay between $7 and $15 for a typical entree, with salads being on the lower end of this range).
Check-in isn't so bad, but it's a little more 'rigorous' than what you're used to. To be safe, I'd recommend showing up at the airport at around 3am to 3:30am (at latest). The whole process takes about an hour and involves the usual fare, but also includes interogations about where you've been, who you've seen, etc etc (it's all for security- and while it may feel strange, it should give you peace of mind knowing that they are this thorough with their security procedures).
Transportation to the aiport, I would recommend taking a train. Taxi is kind of a last resort (costs ~$25-$35, though they'll try to charge you more, because you're going in the middle of the night).