Needs a rule clarification. For example, if doing an any% run can the LA time be used for LA%? Can bike any% be used for any% (double-counting)?

Technically, yes, but because they are separate categories that run the same route, you can only choose one of the available categories to make your time count for (so bike any% can be counted for either either bike any%, or any%).
Stuff like any%, arcade%, and 100% in a single run is possible though, because there isn't much of a reason to do these 3 in separate runs, since 100% is any% and arcade% combined: arcade% assumes a completed career, and the optimal route for 100% is doing career first, so that's any%, and then doing arcade.