Your LL is good for the entire one hour window. For example, if you have a LL time of 9:30am, it's good until 10:30am. Some people have reported being able to use it for up to 2hrs past the time, but that's not guaranteed.
For Little Mermaid, unless they've changed how the allow guests to enter, everyone waits to be seated in the same "holding area." You can enter through the LL and be allowed into the waiting room before guests in the standby line, but everyone mingles in the waiting room. The show runs frequently (maybe every 15-20 minutes), so there's not usually a huge wait there.
Indy has set times. I haven't used a LL there, but I'm assuming LL guests are allowed to enter and choose their seats before the standby line is allowed in. It's a very big theater. If I had a LL for Indy, I'd arrive around 15 minutes before the show. If it's important to you to get a specific seat/area, you could arrive a little earlier, but I wouldn't stress about. There's really not a bad seat in the whole place. Once they allow guests in, the line moves quickly.