Do The Barns have a separate box office?

Yes, at the main entrance.

What are the hours for the box office at the Barns? Or does the Box Office at Filene Center sell tickets for the Barns' shows, too?

Yes

Yes. The Barns has its own box office.

Yes, the Barns has a separate box office

I can confirm Nick's answer. There is a box office just inside the doors of The Barns, completely separate from the Filene Center box office.

When I visited it was on the inside of the facility, so yes.