Bandgap Reference Circuit question
Question 1: The purpose of the circuit is a bandgap regulator, to provide a constant voltage across temperature. (I'm sure you know this, but I want to provide context for other readers.) What is the purpose of R1? You get the constant reference voltage at the top of R1: the voltage across R1 goes up with temperature, and the voltage across Q2 goes down with temperature, so the sum of the two is constant with temperature (achieving the purpose of the bandgap regulator).

In more detail, Q1 is scaled by N relative to Q2, so the base-emitter voltage \$V_{be1}\$ will be smaller than \$V_{be2}\$. The difference \${\Delta}V_{be}\$ is proportional to the absolute temperature (called PTAT) for magic semiconductor reasons. Since the op amp inputs are the same voltage, R2 must make up the difference \${\Delta}V_{be}\$. So R2 must have a current proportional to temperature \$I_{PTAT} = {\Delta}V_{be} / R_2\$ through it. The MOSFET current mirror forces both sides to have the same current so the voltage across R1 is \$I_{PTAT}R_1\$, which we will call \$V_{PTAT}\$.

Now, the voltage \${\Delta}V_{be}\$ across Q2 goes down with absolute temperature, so call this voltage \$V_{CTAT}\$ (complementary to absolute temperature). So the voltage at the top of R1 is \$V_{CTAT} + V_{PTAT}\$. If the resistor ratio is correct, the voltage drop and the voltage rise with temperature cancel out, and you get a constant voltage. This turns out to be approximately the bandgap voltage of silicon (1.22V), giving the regulator its name.

Question 2: equation 6 is \$V_{be}(T) = V_G(T) + \frac{kT}{q} ln[\frac{I_C(T)}{CT^n}]\$. The first term in the sum is the base-emitter voltage at 0 degrees Kelvin, which is the silicon bandgap voltage. The second term is the term that decreases with temperature. The derivation of equation 6 in the paper is unclear to me.

Here's an image from an article I wrote about the 7805 regulator that may help:



The intercept on the left is \$V_G\$, the bandgap voltage. The slope is from the second term in the equation 6 sum. Q1 is the line in orange and Q2 is the line in red. Both lines drop with temperature (CTAT), but the difference between them increases with temperature (PTAT).

Question 3: the capacitor across the op amp is for stability. The handwaving explanation is you don't want the op amp to respond too fast or the system might start oscillating.

Question 4: I'm not positive about biasp and biasc, but I think they are the biases that generate the PTAT and CTAT currents. Thus, they aren't interesting values to compute, just whatever gate voltage the op amp ends up producing to create the desired currents, which are what matters. You could use an equation with the MOSFET properties and the currents to determine biasp and biasc, but I don't think it would tell you anything useful. It's the currents they generate that are important to analyze.