You'd need to talk to her mother and ask if it's ok to use the ring. I'm a huge fan of heirlooms, so I think it's nice. Obviously your relationship with the mother matters, but hopefully it's a good one so she agrees to letting you use it!
For the wedding, you usually exchange bands. Is the grandmothers ring just a simple band? If so, yes, you'd get the ring back and place it on her finger during the ceremony. If it's more of an engagement ring then you would buy a new band (or not if you don't want to) and then use that during the ceremony.
I think it would be. Your girlfriend should find it quite romantic and sentimental that you are using her ring. Perhaps you might want to purchase a separate wedding ring later on so she will have a ring from you and her grandmother. For me, I know my wife would want to keep her parents/grandparents jewellery but not for me to use it to propose. So you really need to either discuss this with your girlfriend or search deep and make a decision.
There is a wedding ring experience where you make your own wedding rings. That might be something you would consider to add more sentimentality. See more info here.