Are patrons expected to stand to let others take their seat? Or will a slight shifting of the legs suffice?
Unless you are especially tiny and we could easily move past you, you should stand up to let others in.
If you have an aisle seat, should you wait until the row fills up before sitting down?
No reason for this, but also, don't get annoyed if you have to get up every 30 seconds between 7:45 to 8pm. When I was seeing Fun Home, I was on the aisle and I stood up to let a woman in who then paused right before she stepped in the row to check her messages. I finally said "we're waiting here" and she said "wow you must be having just an awful day to be so rude" and I said "I just wanted to know if you're coming in the row or not" And then she made me the Enemy of the Row for rest of the show lol.
If you are in a group, should you wait until you are all together to take your seats at the same time?
That is more of an irritation at intermission, when one person in a group leaves to get snacks, and then 2 minutes later somebody else decides to join, then somebody comes back with the snacks, then someone else decides to go to the restroom.......
If you have not checked your coat, is it permissible to hang it on the back of your chair?
This is rude because the seat behind you is not part of your space. That being said, people frequently put their coats on their seatback and onto my lap. Whenever they do, I start to pet their coat, comment on the label, ask my friends to touch the material, I cuddle with it, etc. It's fun.
Can couples put their arms around each other or lean their heads on each other's shoulders?
This interferes with sightlines.
When people stand to let you pass as you make your way through the row, is a "thank you" due? If so, to each person or just a general smile and nod as you go by?
Say thank you. People appreciate a thank you, especially in New York City.
Likewise, if a person passes you without acknowledging your effort to make way, is this rude?
yes it is
Should people check all bags or is that a personal choice?
If you can keep it to within your space, sure, hold onto it, but if not, check it. There was one time when I had been shopping before a show and I had this loud crinkly plastic bag that somehow managed to make a noise every time I inhaled and exhaled (not exaggerating) and it drove everyone around me crazy so I should have checked that.