The question Regression with percentage response variable (ratio of two counts but the counts themselves are not available) in R is very close to a duplicate of What are the issues with using percentage outcome in linear regression?. However, IMO the best (hint to an) answer are @Glen_b comments to the second question. Glen_b also answers the second question, but he doesn't go into some gory details I would be interested in.
In particular, I understand that a logistic GLM is the best choice, but the denominator is not the same among all data points (because students answered to different tests having different total numbers of questions). The suggestion is to condition on the denominator, which would mean making the total numbers of questions an additional predictor, right? How does one deal with a predictor which is not continuous, but it's "more than ordinal"? 20 questions are not only more than 10, they are twice as much, and 3+10 questions is a meaningful operation, etc.
I feel this question is different enough to be asked as a new question. Or should I just wait to see if Glen_b (or someone else) would like to answer the current question? Apologies if this sounds too personalised, but if that's the case, then it's our of esteem and not because I want to pester someone.