Is this mosfet stable? Trouble with capacitances
Yes, such a circuit is always stable.

To make it oscillate you need to change the circuit such that more phase delay is introduced so by:


adding an inductor
adding a 2nd amplifier stage


If you analyze the small signal equivalent of this circuit you should find that it has only one pole and such systems are always stable.

Indeed C1 is in parallel with the Cgs of the NMOS.

If you want to evaluate the influence of Cgd (it's a Miller capacitance!) then analyze this circuit by drawing the small signal equivalent circuit and analyzing it.