The AB200 is wired differently, uses TS jacks, and appears to be set up for switching audio signals. The Boss FS-6 has separate TS jacks for A & B, a combined TRS jack for both A + B switching, plus a polarity switch and momentary / latching switch for each output A and B.
I can't say that the AB 200 would work correctly, and it seems to me that you'd have to create some custom cables at a minimum.
Yes (BK+ and BK- in Bank mode). In Play mode, FS3 would get you Bank Up. Keep in mind that the FS-6 needs a 9V battery / power for the LEDs (it won't switch with it). You might want to defeat the 9V switching at the input jacks, or supply it from your pedalboard. Multiple FS-6's also lock together mechanically.