Just one thing to keep in mind about Monsters Inc is it has really abrupt stops at the loading area. They were significantly more abrupt than I remember from several years ago. So if/when you do ride I'd just make sure your dog is in a safe position where he won't bonk his head or anything when it makes those stops. I'd definitely check out Little Mermaid as others have suggested. It's a fun and gentle ride.

If your dog is okay with rides that have some quick spinning, you should also check out the Spiderman Web Slingers ride and Toy Story Midway Mania. All of these rides have wheelchair access vehicles by the way if that would make it easier to fit your dog on the ride. They may let you do that especially if there aren't any wheelchair users waiting. You may not want to do the wheelchair vehicle with Monsters Inc., though, because the wheelchair area is separate from the companion seat; it's in the front with a little wall separating where the companion sits in the back. With the other rides, though, he'd be sitting next to you.
As far as shows, I highly recommend World of Color if you haven't seen it yet!! It's the nighttime fountain/projection show. For daytime shows, there's a very short but really cool Spiderman stunt show throughout the day right outside the ride. A lot of people really enjoy the Wakanda show, too. I thought it was pretty cool but geared a bit towards kids. It was also really hot when I saw it, though, and the crowd was kind of insane so that may have colored my perception of it! After the warriors enter, they allow the crowd to surge forward to get closer, which I wasn't expecting, so it became difficult then to find a spot where I could see from my wheelchair. So if you have any issues with navigating in crowds that's just something to keep in mind. Since you're not a big Avengers fan anyway, you may want to skip that one, but I say the Spiderman show is worth a watch especially if you're already there for the ride. They do stop showing these around 4:00, though, if I remember from when we were there a couple weeks ago. There's an additional Avengers stunt show that plays in that area, but it wasn't playing when we were there, so I can't give any kind of review of it.
A couple more show options -- there's a shortish jazz show we enjoy called Five and Dime in Carthay Circle. The singer and instrumentalists are great. Then if you're a drum corps fan, towards the other side of the park, there's a fun drum show with the "Green Army Patrol," the Green Army Men from Toy Story.
Hope this helps! Let us know what you think if you end up trying any of these!