Chinese Elm is not an indoor tree. I don't know what Black Spider mite is , I have heard of red spider, but I doubt it is that. Possibly Black Fly. The black tips to the eavs sound like a different problem. I would take it back and demand my money back and get a healthy tree. If you must keep it then a very dilute soapy spray might help. Washing up liquid one drop in the can. Make sure you spray UNDER the leaves.