I don't think the ring had any effect on Gandalf sending the ring towards Mordor. Gandalf did not possess the ring for very long, in fact not at all is about as accurate, and so it could not have attached itself to him. His thought was only to end Sauron's campaign quickly, and the only way was to unmake the ring.
But as far as other's possessing the ring, it would depend on how strong their will was. The ring had a will of its own, we know that. So when someone, or something, puts it on, it becomes a battle of wills. It will be a losing battle for bearer, for the ring will never tire and is from the beginning very strong. Eventually, the ring would overcome anyone. If something has a very weak will, such as a warg or a troll, they would be defeated by the ring quickly and would give it to Sauron. Others might last longer, but in the end they would all become trapped by the ring's power.