Quote: "I guess I could also just add all the characters animations to a single sprite sheet, and just play the relevant frames."
Yes, this option.
I now use Spriter to make animated sprites, which is a much more elegant solution. As well as perfectly smooth movement, you can tween between animations, so going from walk to run is more natural for example.
Spriter Pro often appears in bundles on HumbleBundle.com
Quote: "I need a repeated background. I could make a really super big sprite with its image repeating, but that sounds like something which would take a lot of resources. So I think I will try to go for a cheap hack instead: The background sprite is locked to the camera position. Moving the camera offsets the background sprites image accordingly."
I have a similar situation at the moment. I have created layers, and each layer moves at a different speed to create the classic parallax effect. Each layer is made up of a number of tiles, which I scroll from right to left, and then they recycle back to the right when they have left the screen. I use 4 or 5 different images per layer and randomise the images as they are recycled to prevent the "Scooby Doo" style repetitive background. By doing this over 3 or 4 layers, you get a good enough effect to fool the viewer. I also make some of my tiles reversible, which effectively doubles the number of variations in tile background.
Finally, I add random scenery elements (such as trees), to make it more random and natural. I have a config file that defines which layers the elements are allowed to be placed on, and within what range (vertically) of the layer. Scenery elements allow you to overlap layers to make them "work together" rather than being several distinct strips of horizontal scenery.
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Quidquid latine dictum sit, altum sonatur
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