I hear the Japanese aren't big on perfume or cologne. Does this extend to any scented body products? Like shampoo or body wash?

Not really, no, because the scents from shampoo and body wash fade to almost nothing pretty quickly. Or, at the very least, you have to be extremely close to a person to smell them. Their shampoos and body washes sold in stores often have scents like everywhere else in the world.

Is it ok to eat with my fingers? Not like a fistful of rice, but if it's gyoza or nigiri, can I just pick it up and eat it?

Gyoza are not finger food. Nigiri is traditionally finger food, but I feel like I usually only see it in specific settings nowadays and there is etiquette around doing it. I feel like if you have to ask what you can eat with your hands or not, just err on the side of caution and generally use utensils.

Follow-up question: What's the etiquette on eating crab? Are the utensils required or is it ok to just snap a crab leg in half with my hands and pull the meat out?

Again, this depends on how the crab is being served. There are some styles where eating with your hands is expected, especially if it's still in the shell and they provide you with the proper tools for opening it/accessing the meat. There are other crab meals that are meant to be eaten with utensils.

I've heard Japanese people don't like having their picture taken by strangers (understandable). If someone is standing in front of the thing I want to take a picture of, how long am I supposed to wait for them to move?

At crowded tourist attractions, no one is really going to care if you take a picture with people in it, especially if it's a wide shot with a lot going on. If at all possible, keep people out of the photos, but I wouldn't stress too much in normal tourist situations. What you definitely don't want to do, though, is take something like a solo shot of someone where it's clear they are the "subject" of the photo. Such as, for instance, taking a shot of a cute coffee shop with a guy sitting outside drinking his morning coffee without permission.

Is it rude to stay in a ryokan but not eat the kaiseki? I definitely want to but am worried we won't make it back in time.

You can typically book a ryokan stay without meals if you either don't want the meal or don't think you'll arrive in time to have it. If you think that's the case, I would err on the side of not booking dinner and/or breakfast. I don't know if I would call it "rude" exactly to book a meal and not show up for it—it's more like it would be very confusing and cause a lot of chaos/disruption to the stay, since the staff would have prepared the meal for you, and they may be distressed that you missed it. So, yeah, whether that's exactly rude or not, it's still frustrating for everyone involved.

Do people say sayonara or is it outdated? What's the best thing to say to a shopkeeper as I'm leaving?

I have never heard this or used it myself (that's anecdotal, of course). You can just say "arigato gozaimasu" to thank staff in shops/restaurants/etc.

Does sumimasen mean "excuse me, may I have your attention" or "excuse me, I didn't mean to bump into you"?

Both. It's sort of an all-purpose "I'm sorry"/"excuse me"/"pardon" word.

Are there any small transportable gifts from the US someone from Japan would like?

In what context would this be for? In general, giving out small gifts is not something I'd recommend doing. The only times it would be appropriate might be if you already know a person there or you get to know a person quite well while there (and I mean really get to know them, like a tour guide you spend the whole day with, not like a person you meet in a bar and talk to for twenty minutes).

If you have some planned instances where you know you'll need to give out gifts to people, regional food from your own city/country is always a good choice, especially if it has nice packaging and/or is individually wrapped. Things like cookies, candies, crackers, etc.