I'm really lost when it comes to amps, effects, and the likes, so I have a presumably simple question. Would it be possible to run a multi-effect board like a Boss ME-70 right into a PA? Would it sound ok?
yes and if you tweek the settings right it would sound fine.
Me, one of the guitarists, and the other bassist at my church do it all the time. It sounds alright. Would be better with an amp, but it gets the job done.
The Boss ME-70 doesn't have cab simulations. I would recommend getting a multifx that has cab simulations for hooking up to a PA. Cab simulations are specifically designed to allow you to do this.