Not that it’s a bad idea to have multi-move chess as a playable variant on chess.com, but “Chess sucks without something that can (effectively) teleport” seems to be the theme of your suggestions.
Variants
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These variants are not my idea, they were others ideas I just like the idea of the variants.
Unfortunately, it takes a lot of effort to add variants with completely different rulesets to the website, but we appreciate your suggestions :)
You might be able to create a similar variant to the third one using the "Taboo" rule. (eg no checks) but I don't have much experience with creation of variants.
Progressive and double move chess is a really cool concept but extremely hard to realise on the website. Would require staff to come and add coding.
Are the mods here able to add new chess variants?
I have some ideas:
Penguin Chess: Every piece can move to every empty square. Capturing is like in normal chess.
Pawns can't move to 1'st and 8'th rank.
King can only move legally.
Aim is to checkmate.
https://en.wikipedia.org/wiki/Marseillais_chess the second one
Third one: Same as normal chess, but no checks only checkmate is allowed.
Progressive chess: https://en.wikipedia.org/wiki/Progressive_chess