1. It depends. If you have been handed the regular menu, go for it, with one caveat. I have been to dinners where money isn’t a factor. However, there’s often one menu item at a restaurant that “stands out”, such as a giant steak or ice cream topped in gold flakes or something. These are usually items that people order strictly to show off, so if you’re eating on your host’s dime then I wouldn’t ever order one of the “showstoppers” because it seems pompous and crass, especially since there’s usually some fanfare involved by the staff when they bring it to the table, like five knives being stuck straight up in the steak or the gold-flaked ice cream coming out in a giant, shiny goblet. Also, please, please don’t order something “expensive” and then take pictures of it to show your friends. >.<
If you have been handed an abbreviated menu and there is only one “expensive” item on it, I would hesitate to order it, especially if you know the hosts are on a budget.
2. If your hosts are drinking it’s fine. Otherwise, and especially if no one else is, I wouldn’t.
3. No. They put it out there for a reason—if they didn’t want to pay for it, then it wouldn’t be in the bar.
4. With #3 being said, one of the only rudeness issues I can see with this is how many drinks it would take to make you drunk/inappropriately tipsy. Again, smart hosts know what their budget is and they have planned accordingly.
My answers wouldn’t change in terms of who was hosting the event, they just change based on what is happening at my table. If everyone’s ordering the moderately-or-low-priced items and not drinking, that’s what I go for.
Also, even if the hosts know your particular “habits”, it’s not really their places to notice if you’re drinking, unless you’re the only one drinking or you’ve decided to get schmammered at the table. Ditto for what you’re eating, unless you’re doing one of the things I’ve mentioned above, like ordering a “showstopper” or the “one expensive option” or something that’s $10 higher than what everyone else is having.