It’s is not, it isn’t ain’t, and it’s it’s, not its, if you mean it is. If you don’t, it’s its. Then too, it’s hers. It isn’t her’s. It isn’t our’s either. It’s ours, and likewise yours and theirs.
If you are using this as a mnemonic about its/it's, this is fine; but your statement of it is as a rule ("the apostrophe replaces the missing letters in contractions") is misleadingly incomplete. Apostrophes do that, but they also serve as a possessive marker in the general case ("Sam's"), which is of course why its/it's causes so much trouble in the first place.
No need; I think the original formulation---as a mnemonic---is just fine. We don't really need a "general rule" here anyway, and to be honest, English orthography and "general rules" don't really go well together.
The only reason I posted at all is because linguistics is an area where a lot of quite intelligent people hold some extremely unexamined (and incorrect) beliefs, and there is furthermore a common tendency to propagate those beliefs as if they were fact. As a result, whenever I see someone articulating anything that is formulated like a general rule about a language (or about language in general), I try to make corrections where I can.
You took "the apostrophe replaces the missing letters in contractions" as me postulating that all apostrophes are only used for contractions, which is jumping to conclusions, IMO.
Even if that statement is taken as a general rule, I can't think of any contractions that don't use an apostrophe, and it certainly doesn't state that contractions are the only place where apostrophes are used.
