Well in English, when you want to pluralize something it depends on the type of word and its roots. But for acronyms and abbreviations you just slap an S on the end. Unless the word ends in an s in which case you slap es on the end. So presumably it would just be TARDISes.
Consequently, that explanation also answers the second point.
If it were a word on its own, sure, but it's been repeatedly shown to be an acronym and in English we pluralize them but also treat all acronyms as if they're singular unless an s is added. That and sheep/fish are exceptions to the rule, not so much the rule itself.
Did you know fishes is an acceptable way to pluralize fish? Like, not acceptable to me, but is apparently agreed upon in language? Sounds so fucking bizarre... especially as a dude from an island obsessed with fish.
The jokester in me wants to say TARDIS â TARDES by analogy of penis â penes but the fact of the matter is that Time and Relative Dimensions in Space doesn't really pluralize even when it isn't abbreviated