I use Alieene's tacky glue or Elmer's all purpose glue. For high gloss those usually work but for more difficult ones, I use a hot glue gun.

For the other issue, if your cardstock tears a lot, be sure to check your blade that it is not chipped. If it keeps cutting wonky, put in a new blade. Another issue is if you have bits of fluff all over your matt, either clean it off properly or use a new matt.

I use 110# index cardstock for most needs Smooth 200 gsm.

I hope you let us know how you get on with it all.
Happy cuttitng