if the surfaces are mated perfectly, and properly prepared(bare wood), just hide glue should hold the bridge in place. but often screws(bolts) are used on cheaper guitars. screws/bolts are not ideal because they add a lot of extra weight...

the bridge should glue after soundboard finishing or before finishing?

You can put the bridge on either before or after finish. I place it, then tape off the bridge position and then finish. Take the tape off the area of where the bridge will be, and glue it on. I do use vacuum press for clamping the bridge.

Now as to adhesive. After sanding the radius , you can do whatever you want as adhesive. I am not sure about CA. I haven't used it or paid any attention to post on those that do. I have used LMI stuff, Titebond and Gorilla wood glue and never have had a problem with any of them, whether it be braces, binding (I use CA for this a bunch) top and backs or bridges. I am sure by most HHG would be considered the best, I just never use it, and so far no need to. That includes two higher end shops I have worked in. Not to say I won't try it sometime, just not interested in messing with it now.

Traditionally, the bridge is glued first on classicals, and finish first on steel strings. Mainly due to the finishing methods used (French polish versus sprayed finishes). I French polish steel strings too, but still finish first because I like to use fancy bridge shapes that wouldn't be possible to polish around perfectly.