Given a t-SNE plot, how can I infer the "most correct" labels? How does one understand its structure?
With t-SNE none of the input parameters are weighted more than any other parameter so the differences you want to see like students forming islands by grade level will not happen because there is so much other data present to pull those students/data points in different directions. 

I highly encourage you to have a specific question in mind and tailor your input categories so that your question can be answered by the t-sne map.

You might try asking a specific question and changing which input categories you ask t-SNE to look at. For example, do taller students get better grades? Feed in height and grades  categories while leaving out grade level and age. This is a silly example but I hope it gives an idea of how you can use t-sne to help you learn about your data. 

You might also find that there are categories that mask meaningful findings. Height might not be very useful for pulling out meaningful information and since the height range is going to be much larger than the A-F grade range, it will likely influence the t-sne map more. 

Looking at the data as you described on a color scale for each category is a good place to start with each new t-sne run. 

It is okay to run multiple t-snes with different parameters. To be sure you are getting a meaningful answer, I recommend running t-sne multiple times with the same parameters as well. 

If you have enough students, it would be great to included half in your training set where you explore and figure out what questions to ask and then when you think you've found something meaningful, apply those conditions to the other students in the testing set to see if it holds true.