Drip under car caused by clutch or something else?

Defrost settings will run the air conditioning and could cause condensation drips from the evaporator core. This would be a water drip from the firewall area. The only thing you would really have to worry about damaging while teaching a new driver to drive a manual transmission is the clutch disc itself, unless you are trying to teach her to drive like a trucker (the statement about shifting into first while using the accelerator has me concerned about this. There is no need to teach a novice that kind of stuff. Just drive normally, use the clutch for shifting, and leave the synchro-shifting to the truckers). Main thing is, though, this is a new car with 200 miles on it. If you are worried about it, take it back to the dealership and have their service department look at it.