First of all is it really possible to have a nmos with negative threshold voltage? If threshold voltage for M1 was positive, then M1 would be in cutoff. How do I analyze this circuit?
First of all is it really possible to have a nmos with negative threshold voltage?
Yes, it's called a depletion mode MOSFET and, in your diagram, the MOSFET symbol is a depletion mode MOSFET: -
As wired in your schematic, M1 (being a depletion type MOSFET) is acting as a constant current source.
Picture from this document entitled "An introduction to Depletion-mode MOSFETs".
An example of a depletion mode MOSFET is the BSS126.