Proper method of connecting to vent

In my basement, I have 4" of room below the bottom of the floor joist to connect to a 2" copper vent pipe. I have a few questions:
1. How should I secure the vent pipe before I cut into it?
2. What type of fitting should I use?
3. How should I transition from PVC to the copper?

I'm venting a Drainosaur that has a 2" vent pipe.

Thank you for help with some basics.

If you are describing that pipe with a cleanout cap, are you sure that isn't a drain because you cannot tie into a drain to vent.