Would these machines have been networked with CAT-3? Daisy chained phone cords?

More likely coax. 3com 509c network cards. Much less infrastructure to have a lan that way.

IBM had a network that ran over phone cords that were daisycbained from one node to the next.

It's also possible they used coax, either ethernet (10base2) or Arcnet.

Depends on the exact date.

I never worked with DOS BBS systems, so I can't say about this photo specifically, but the ones I did work with had between one and four dialup modems hooked up to each machine, depending on its capabilities. They did "networking" through a store-and-forward messaging system. It wasn't networking as we'd recognize it today.