Well, can't really answer your question about the transmission, but at first instict it sounds mechanical. Sometimes a good tranny fluid change will do wonders. Search this board for "magic reset" and see if that thread helps at all.
The second question about your coolant: the most common problem is a faulty level sender. Not sure where it's located but it basically get's stuck and sends the wrong signal or starts reading a lower level than what's actually there in which case it has failed. I hear it's not too expensive of a part to replace, but I don't know and that's all I can suggest.
Hope it helps.