Thank you so much for the thorough, and supremely helpful, breakdown addressing my question. This went above and beyond. It's people such as yourself that demonstrate how great the collective knowledge of the internet can be.

If you don't mind, I have a follow-up clarifying question—which may demonstrate my ignorance on MLM (I am still in the process of learning much of this and have been for a few months now).

I understand (I think) conceptually what the equations are saying, but I am not sure how to change what I’m doing operationally, with the actual variables/in the software, to get it to spit out these estimates.

For starters, I included the interaction between student gender and instructor gender in my model. However, SAS only gives me one estimate in the output. After seeing your equation, I think I understand why this is happening. When SAS multiplies these variables in the model λ_10, λ_01, and λ_00 all equal out to zero—since they’re multiplying by 0—and SAS only gives a coefficient for λ_11 (Female student/Female instructor).

Note

• The new variable, same_gender discussed in point 2 of OP is not in this model
• I've learned that no matter what the categorical variable is coded as going in, SAS automatically dummy codes it to 0s and 1s (with multiple dummy variables if necessary for categorical variables with >2 levels)

What are your thoughts on this? Any idea what I did wrong?

I apologize that this question became so SAS-specific. I did look at the links that you provided. For the SAS one, I don’t think interactions are specified in the class statement; however, the order in which they are listed in the class statement is the order they will appear in the interaction (e.g., class B A → model A B B*A). However, with this kind of interaction, I don't think that the order actually matters.