If your supervisor said that you should include a variable like that in your model because "that's what people do", she should resign. If someone did that, they should be banned from all journals for life.

Too lazy to do the test but looks like they're not. Definitely not close to a $10^{-10}$ p-value

Your threshold strictly determines FPR and TPR so it wouldn't add much new information and it'd make the chart more harder to interpret

If you have no further information, there's no way. You'd need an infinitely-sized sample for that. However if you are working with a limited set of options (for example, if you know your data follows some normal distribution), then you can use your data to narrow the options down

@Peter Sure! There's no limitation in that regard

Does this help?: ncss-wpengine.netdna-ssl.com/wp-content/themes/ncss/pdf/…

You can get a quick estimation by calculating the average height of the plot. Calculate FPR for $TPR=0$, $TPR=0.01$, $TPR=0.02$ and so on until $TPR=1$. Its average is a good estimate of the area under the curve. For something more precise, take a look at some numeric integration methods. I've editted the answer to deal with this part too. I hope it helps.

Let's say a car is on the road at 60km/h. It's position (as a function of time) is a non-stochastic variable that changes 60km every hour and therefore, not a constant. Its speed is a non-stochastic variable and also a constant.

@LeonidMednikov why not do a million different test for a million different possible distribution and pick the lowest of those?