Can is use the 2N7000 MOSFET to switch 12v from 3.3v?
As JonRB said, NO. 

The parameter to look for in a MOSFET datasheet is the Vgs at which the transistor has an acceptable Rds (series resistance) and can conduct an acceptable current. For your 2N7000 the relevant part is on page 2:



The first line for the Static Drain-Source On-Resistance is for 10V at the gate, which is for 500mA, which sounds OK*. The second line, for 4.5V at the gate, might still be OK for some purposes, but not for your solenoid, which draws 120mA. Note that there is no line for a lower Vgs.

*Never put your trust on a single entry in the datasheet. At 500mA with an Rds of 5 Ohm, the dissipation is 1.25 W, which is far above the 350 mW stated on the first page**. That 500mA/1.25W is probably OK under some very specific circumstances, like keeping the casing at 0 degrees C.

**Never trust a figure on the first page of a datasheet, read it as if it were written by the marketing department, not the engineers. But you can trust that a device will never do better than what is stated on the first page.