Every aircraft that has TCAS II installed now, has Mode S and I'm sure about that one.
Indeed, you're correct in your scenario. If there's no altitude reporting on the target, you only get a TA (Traffic Advisory) with the Traffic call. Look out and ask ATC.
If you get an RA (Resolution Advisory), the aircraft will issue you a vertical pitch command in order to stay clear of the traffic and the command will be announced (eg. "Climb", "Descend now", "Increase vertical speed"). That command must be followed unless you're absolutely sure that the other traffic is no factor.