Mermaid Lagoon Theater and Magic Lamp Theater have Fastpasses. They're the continuously run type shows, actually listed under attractions (instead of shows/parades) on the TDR website, so not sure if that's what OP was asking. We enjoyed the show at Mermaid Lagoon a lot, thought it was pretty different from anything we've seen in the U.S. parks. It has Ariel "swimming" above hanging by wires and a giant animatronic Triton. It's a circular theater and Ariel goes all the way around, so it felt pretty immersive. Magic Lamp Theater I thought we missed out a lot not speaking Japanese, so it was more forgettable to me. I think there's a translation device you can ask for, but we didn't try it.
For the lottery shows, TDL had One Man's Dream II and priority seating for the castle projection show. We had no luck with the lottery our first day, then the second day was quiet enough that there was no lottery. We lined up for One Man's Dream II about 30 minutes before and got in. It's a lot of dancing with different characters, sets, costumes - not really sure if there was a plot. I thought it was good, but not really a "must see".
As mentioned already, TDS has Big Band Beat, which is supposed to be great, but we weren't lucky with the lottery. I've read that the first show is all standby, but we didn't try it.
We also saw the outdoor show in front of the S.S. Columbia at American Waterfront in DisneySea. It's a different show now, but wanted to mention that there doesn't seem to be much shade, at least there wasn't where we were sitting and it was very hot (in June) with the sun beating down!
We saw the Duffy show at DisneySea. You have to eat at Cape Cod Cook-off (counter service restaurant) to see it. We went in the afternoon for drinks and dessert and there was no line (on a not particularly busy day). My 4 year old DD enjoyed it, the rest of us enjoyed the aircon and a place to sit.