I have a decent sized cage and would like to put a real plant in there for her to climb on. That being said I also have a dog. what is dog safe and glider friendly that would also be sturdy.

Putting a plant inside the cage is not usually recommended. Most potted plants have chemicals and fertilizer pellets in the soil that you would not want your glider consuming. YOu would probably find that the dirt will get thrown all over the cage - and the floor around it if you did put a plant in the cage.

You can use artificial plants as long as they do not have wires in them. I use the plastic vines from Michaels and just drape them over the cage. My gliders like to pull the leaves off and carry them to their pouch or into their tunnels.

you could cut a branch from a glider safe tree like a crab apple tree if you know that the tree has not been sprayed for fungus or insects and as long as you clean it well(scrub and let dry completely).

Your glider will mark it with pee, strip the bark and leaves from it any ways....and it will stink eventually and you will have to throw it away.