Does my puppy need vaccinations even if I keep them inside?

Yeah. We don't live in a bubble. Even in this world of coronavirus, we don't live in a bubble. We are going to be exposed to airborne pathogens, things like that.
Parvo, for example. If your dog never goes outside, not to say that's great, but okay if that's your choice, but parvo can still find its way inside. Why? You go outside. What happens if you are around another dog and there are viral particles on that dog? Maybe they have a bowel movement or had a bowel movement that you didn't see, you step in it, it's on your shoes... something like that. If that viral particle makes its way back into your house, well, guess who's going to find it? Your dog will. Is it as common in indoor dogs? No, of course not, but it's still possible. And once again, as I said, an ounce of prevention is worth a pound of cure, especially when you're dealing with potentially fatal viruses like parvo.

I would say absolutely. By law, the core vaccines DATPP and rabies should be given. But I also recommend the other ones that I mentioned, such as lepto to prevent kennel cough.

Absolutely. Rabies is required by law. It doesn't matter if they're inside or outside. That's a given. They have to get the rabies vaccine. There's also distemper parvo. Even though your dog never goes outside, you can potentially bring in these viruses. How do you do that? Primarily through like the soles of your shoes. You can pick up the organisms in the environment in the backyard. You go to a neighbor's house, to a park, you go anywhere. Especially if you go to like a dog park, you can easily pick up some of these viruses and bring them back to your dog, even though your dog never goes outside.