Probably depends on the bioweapon. For instance, I'd expect it to be effective against anthrax spores, which was the most recent bioweapons scare we had.
You're still using outside air, but by maintaining positive pressure you ensure that it's only coming in through the filter, rather than a mix of air intake and leaky joints.
Not sure about particular type of filter, but CBRN countermeasures found on tanks, APCs and such are also basically air filtration with positive internal pressure.
It absolutely is, but that doesn't mean it wouldn't work in principle.
A HEPA filter is used in a lot of labs where biological substances are handled, even potentially hazardous ones. It should filter many types of biological agent out of the incoming air, and the positive pressure will keep such things entering other ways.
I think the reason they put it in is so people would talk about it and, look, we are. It is a marketing gimmick that likely doesn't require any additional hardware in the car (the air pump and HEPA filter are already in it, this just turns the pump up and adds a new dash button).
I'll give Tesla the nod for one thing: They sure know how to market. Between the wing doors, the bio-protection mode (which will be on the nightly news), and insane mode they're very savvy.
You're still using outside air, but by maintaining positive pressure you ensure that it's only coming in through the filter, rather than a mix of air intake and leaky joints.
This will be used 99% for skunks.