I am using a solid green spider plant in my vivarium right now. I took precautions so that it wouldn't overtake the drainage layer with its roots. I wrapped them in carbon-fiber mesh and a rubber band and then covered them with soil. It should be mentioned that my spider plant had approximately 1 cm long roots to start off, so I wasn't too worried about the roots. The plant itself has only grown about 1 inch in about 8 months, so, I am also not worried about the plant getting too big. If it all of a sudden shoots up or the roots start growing like crazy, I will remove it from my vivarium.
I think that small spider plants can do well in a vivarium. IMO, the solid green ones are more aesthetically pleasing and look more natural.
I don't, but I have had it in my glass house before and it took over as a ground cover and outcompeted some of my plants. It's quite the thug. If you want a better behaved alternative I have used green dwarf mondo grass in tropical terrariums before and it was pretty good. It looks quite similar to the green spider plant but is smaller and isn't as aggressive.