Is there a "what" that is sort of universal, that we can make our go-to word for "what", if we can't remember anything else?
Not really. Translations EN<=>PL are not 1:1.. There are some structural differences that make it hard.
No, no shortcut for that, I'm afraid.
In conversation co? could be probably understood for some declension of co? Probably understandable , not ok.
Jaki? will be understood when used instead of other genders of cases.
But jaki? and co? are as different as how? and who?
'what's your number' easily the best translation in English