Will a 1.5gb/s connector work with a 3gb/s slot or the other way around?
When you say "connector" do you mean the data cable? If so, yes they will work for all standards of SATA. Some older SATA cables do not have the metal clips that hold them in, but they will still work.
Alright what if my hd is 3.0gbs and my mobo only supports 1.5gbs?
HD's are backward compatible. You'll be fine. Some of the HD's sometimes have a jumper to limit the bus speed, but I never had any problems connecting 3Gb/s drive to a 1.5 Gb/s socket.