Is the adiabatic flame temperature equal to the nozzle stagnation temperature
The adiabatic flame temperature is definitionally the lossless temperature of product gases following a combustion reaction. So if you assume there are no heat transfer losses in the chamber and that the temperature profile is homogeneous (and that you have complete combustion!), then the stagnation (total) temperature of your nozzle flow will, indeed, be equal to your adiabatic flame temperature.
Of course, these assumptions are not precisely true. There IS heat transfer into the chamber walls, the temperature profile of the products is NOT homogenous, and there are viscous and turbulent losses as well. Heat transfer is the most pronounced of these losses. It is typically neglected in simple analyses because heat transfer in Watts is usually much smaller than the power of the combustion products. However, the effects have been studied.
Combustion performance in a rocket engine is characterized by a parameter called Cstar. This is the characteristic velocity. It is purely a function of combustion product specific heat ratio and total temperature. It also directly scales an engines specific impulse, so combustion designers desire to maximize Cstar.
The losses I have described are captured in Cstar efficiency which relates in ratio the measured Cstar for an engine and the ideal one you would calculate using thermochemistry.
I have seen values for this efficiency ranging from 50% to 95%, so it really depends on your engine. For most basic analyses, you'd be safe assuming a 100% efficiency. Factors that drive down cstar efficiency will be high rates of cooling to the chamber, incomplete combustion, very short duration fires, etc. Anything that deprives the flow of the assumptions I mentioned at the beginning of this answer.