ABS Yellow glue by Oatey's is by far the best cement for ABS. I have tried many things and using epoxy is not as good as a solvent bond, which is literally melting the components together.
No, most plastics are smooth and non-porous. For wood glue to work well it needs to soak into the materials that it's gluing. It can not do this with plastic, so will seem to stick at first, but will come loose when fully cured.
Sticky gue
Epoxy.
I have used PL200 and 100 % silicone to do this. Both worked well
It's hard to guess, as you haven't said WHAT you are gluing plastic buttons onto.
Join the plastic bottles with tape and then stick it to the cardboard by sticking chart on it and then gluing it to the cardboard.
Sewing buttons on is better than gluing them.
It depends on who is doing the gluing, and on the use of the two boards after joining together. If the job is one where the joint is to be visible, gluing is aesthetically better. If I was gluing it, my joint would be infinitely stronger than one screw. -However someone else's joint may NOT be as strong as one screw.
Depends on what you are gluing. For a flexible material, urethane adhesives are a good bet. For thermoplastics and metals, thermal welding (or solvent welding, for plastics) is preferrable. Don't forget that there are many grades of epoxy, and some are many, many times stronger in bond strength than hardware-store types.
CNC cutting has a very wide range of applications, like cutting metals and plastics, but also sawing, drilling holes, gluing, welding, fiber placement and bending.
On the most bottles of Super Glue it says under the Directions label "Do not use on Fabric" so I would think that wouldn't work too well.
Depends what you are gluing it onto - more detail needed please.