In this regulated cascode current mirror how do we find the minimum voltage across the current mirror? Also, I have read that the current mirror will work even if M1 goes into linear region. But how does it go to the linear region in the first place and how does the current mirror function when M1 is in the linear region?
Look up how a cascoded mirror works, M5 and M2 are the mirror, M1 is the cascode.
The cascode is used to keep the Vds of M2 somewhat constant. M3 forms a local feedback loop to make the cascode's gate voltage (node X).
When Vo gets lower at some point M1 will have such a small Vds that it will enter its linear region.
M3 makes sure (through the feedback) that Vds of M2 is Vgs (of M3) so like a diode switched MOSFET, it must be in saturation.
M1 is not essential for the operation of the mirror itself because which transistors form the actual current mirror ? (I wrote that above).