First question: if the franchise gets a sequel, what should it do differently?
The biggest thing is that it needs to find and stick to a tone. Movies are allowed to be emotionally dynamic, but there needs to be something which brings it all together. Tonto's back story is particularly gruesome, and in a movie with silly fight scenes and jokes about Native Americans it's impossible to tell if we're supposed to laugh or cry. I personally thought the light-hearted and comic side worked better, but it could certainly go the other way. Dark comic book heroes are fine and have worked well recently, but if you choose that route you can't cast Captain Jack Sparrow as Tonto. Johnny Depp can play serious roles, but here his character is primarily physical comedy which just does not work next to heavy issues like the Native American genocide. Pick a side and stick to it.
On the other hand, something that I think really worked for the film which it should stick to in a theoretical sequel was its position with regard to the original story. I'm no Lone Ranger expert, but this movie doesn't exactly hide its attitude toward its source material. We get the famous theme song played over the big action scene, but beyond that Tonto consistently treats with disdain the Ranger's attempts to bring back iconic items from the mythology. The Ranger shouts "Hi ho Silver, away!" and Tonto's reply is a deadpan "Never say that again." The story is told from Tonto's perspective with the Ranger relegated to a supporting role as a bumbling buffoon. This revisionism is the right strategy to take when your source material would have trouble making it in modern society (see: racism) and its mockery of the hero is simultaneously funny and compelling. For more about this issue check out this amazing article over at Jacobin which is better than anything I've ever written.
Second question: what did The Lone Ranger do to earn its Visual Effects and Makeup and Hairstyling Oscar nominations?
The Lone Ranger's special effects are certainly interesting. Watching the movie you can tell which scenes are shot on location and which are shot on a sound stage, but all the action sequences do look really nice. The reason for this is that they used multiple layers of green screen images (they actually used blue screens for The Lone Ranger but you know what I mean). So here's what happened: the actors were shot in front of a green screen, but instead of inserting a single moving picture beneath the actor (which can create depth issues if the camera angle changes too much) the green is replaced with a dynamic layered mosaic of secondary elements which allows real and natural camera movements. If you're curious check out this video for more information. Other movies have done this in the past however, and I don't know if what they made here is quite enough to beat out Gravity for the Oscar.
The Lone Ranger was also nominated for Makeup and Hairstyling, and while this may seem surprising it is pretty justified. William Fichtner is almost unrecognizable as Butch Cavendish and Johnny Depp's aged Tonto is rather remarkable. I don't know if it's enough to beat out Dallas Buyers Club (it's hard to distinguish between the actors' makeup and their bodily transformations) or Jackass Presents: Bad Grandpa (which succeeded in making Johnny Knoxville unrecognizable both on camera and to the unsuspecting subjects of their pranks), but it felt like it deserved to be part of the competition.
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Would I recommend this movie? Probably not. It's too long, and even if it were trimmed down the shifts in tone are hard to accommodate. But it is undeniably an interesting film, and while I don't plan on watching it again any time soon I don't exactly regret seeing it. There is a lot in it which is worth thinking about which ought to be worth something, but it is more fun to discuss than it is to watch.