Hi,
no worries.
1) A bit cryptic slide... g is the voltage gain not the transconductance. Given the diagram and what they mention in the first sentence they assume g = -1 and a PWL function. That gives you the straight lines and the VOL = 0 and VOH = VDD, and VM = (VOH+VOL)/2 = (VDD - 0)/2 = VDD/2
2) Haven't really looked into the details, but it should be possible to derive directly from something like:
g = (gmp + gmn) / (gdsp + gdsn) = (2 ID / Veffp + 2 ID / Veffn) / ( lambdaP ID + lambdaN ID) = (1 / (VDD-VM-VTP) + 1 / ( VM - VTN) ) / ( ( lambdaP + lambdaN) / 2) = (VM - VTN + VDD - VM - VTP ) / ( (VDD-VM-VTP) * ( VM - VTN) * ( lambdaP + lambdaN) / 2) = ( VDD - VTN - VTP ) / ( (VDD - VM - VTP) * ( VM - VTN) * ( lambdaP + lambdaN) / 2) etc.
In this case, I think they have made some assumptions, that the gm of the N and P are equal, and that the r factor might be some correction factor for that. I you do those assumptions, one can land at something like the expression above.