Regarding your question "what is it?", verdigris is apparently a complex mixture of copper compounds. Carbonates, sulphates, sulphides, oxides, hydroxides, anything odd like that in the air or environment can get sucked into it.
Yes, it will continue to "take over" the coin unless you seal it in an absolutely airtight container and never take it out, and yes it is "contagious" - any other copper/bronze coin kept in contact with it for awhile will start to turn green too.
Don't say "infinitely" when you mean "very"; otherwise, you'll have no word left when you want to talk about something really infinite. - C. S. Lewis