Just to preface - most people here (including myself) don't do in-person betting. Online is much easier. But having gone to Vegas a few times recently, I do know a little about it.
1. I don't know if there's an "easiest" casino, but the process for all of them should be pretty straightforward. Sportsbook areas are rarely crowded tbh. You shouldn't have much trouble. I was at Mandalay and Luxor last weekend during 227 and there was hardly anyone at those areas lol. The "do-nots" I would say would just be general sportsbetting tips. Don't waste money on huge favorites, don't get too caught up with parlays, etc.
2. They do. But the odds are common between sister casinos. For example, MGM operates Bellagio, Mirage, Mandalay Bay, and a couple others. You can see the originators for each casino
here to see which casinos share what. That same website gives odds between the different operators (they'll show when we're closer to an event, like
here )
3. Depends on the casino. Can be 30 days, can be 120 days. Check with the casino where you place your bet. I don't think there's a way to insure your slip specifically, but you can take a picture of it. If you lose it, you can give the casino information about it so they can find it in their system. Otherwise, you can purchase a player's card, which can make it a lot more secure for you. You can apparently earn perks with them too, but I've never used one myself.