Former itamae here. For context, I’m Japanese American (born and raised in the US), and worked behind the bar for about five years.
There’s a lot of good information being shared. I think many of your questions will tend to take care of themselves depending on what kind of place you choose. If they have a menu with a jillion variations of Monster Truck Rolls and there’s at least one table of Sushi Bros pounding sake bombs, you’re not getting a traditional sushi restaurant experience, so do whatever you want. If, OTOH, the place has a case full of fresh fish at the bar and the itamae’s hands look like he (or she) could pound nails with them, your expectations should be heightened. Omakase is not a bad way to go. It allows the itamae to choose from what they’ve got on the day that s/he thinks you’ll like. It’s not usually a set course because the selection varies with seasonality and availability of fish. Or maybe they’re just sold out of something. It’s perfectly ok to tell the itamae if you have any dislikes. I, for instance, think ika (squid) is a waste of money and effort. I also won’t eat tako (octopus) not because I don’t like it, but because they are so smart it bothers me to eat them. (Yeah, I still enjoy pork in all it’s forms - so sue me!) Most importantly, though, is to tell the itamae that have a little experience with sushi and you’re expanding your knowledge base, then provide some feedback on what you’ve been served. “I didn’t like this, I loved that, do you have anything else that is similar?” goes a long way to getting the best experience possible. Also important is to give the itamae how much you’re willing to spend, lest you get sticker shock when the bill arrives.
Personally, I prefer to start with milder items, and build toward stronger tasting items. Typically among the last things I’ll have are saba (pickled makerel), kohada (pickled gizzard shad), or something similar. My very last, assuming I’m not already stuffed, is tamago (egg/omelet), which is lightly sweet and a little salty, too. My favorites are toro (tuna belly), hotategai (scallops, also sometimes called kaibashira), and hamachi/buri (yellowtail).
As for whether the restaurant will mind seating a non-sushi eater at the bar, just ask. If they’re busy, they may not allow it. As long as your companion eats some sushi, though, they probably will be allowed to order non-sushi items to eat at the bar, too.
ETA: If you really want the best possible experience, go on a night when the place is not getting slammed, chat up the itamae, and buy him a beer (or other beverage). Some of my best customers eventually learned what I liked to drink, and would stop by the cocktail bar to buy one for me before sitting down in front of me.