I really don't know if there's a MOSFET that can be properly controlled with 3.3V. The threshold voltage is where it starts to conduct. You want to "slam" it on (saturate it).
A regular (bipolar) transistor might work better or the MOSFET might need a transistor driving it. (A transistor turns-on with less than 1V but you generally get more voltage drop/loss across a transistor which means it gets hotter and can burn-up easier.)
No, it should turn off when the gate is grounded. You didn't show us your schematic. Does your circuit have a flyback diode? When you disconnect or turn-off an inductive load there is a high-voltage "kickback" that can fry a MOSFET or transistor.