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What do the different sizes represent?

Tiny is the equivalent of 148x197 pixels. Small is the equivalent of 184x229 pixels. Medium is the equivalent of 220x261 pixels. Large is the equivalent of 256x293 pixels. Very Large is the equivalent of 310x341 pixels

Are there any other customizations that I can make to the puzzles?

No. Unfortunately, the appearance of the puzzles are controlled completely by shredderchess.com so you can't customize the colors or board in any way at this time.