If I have a bike that belongs to someone else but the registration is not current, and I am able to pay registration fees online (since no ID is required), would this meet the Team Strange rule for a currently registered bike?

I am not a TS official, I didn't even play one on TV. That said, others have ridden borrowed bikes in TS events. Riding a bike not registered to you might be a bigger issue with Johnny Law and that might hurt your score.

I'm also not an official anything, but my take would be that it would depend on the rally.

On the MN1k, lots of people have run borrowed bikes (or in a couple cases, rented bikes!). On something larger, like the ButtLite, where they check insurance papers to the VIN number on the bike, you might run into some roadblocks.