What's the difference between 学 and 学习?
Like many Chinese words, 学 and 学习 differ primarily not in meaning but in length; in many cases, 学习 is chosen over 学 because the sentence calls for a disyllabic verb for reasons of prosody. 

Your example number 2 is a good one. We say 我在大学学习 not because 学习 can be used intransitively but 学 cannot be, but because the sentence *我在大学学 sounds "incomplete." (You may have already learned that many disyllabic verbs cannot be used with monosyllabic objects; this is just one example of how prosody often governs the formation of Chinese sentences.)

Many (most? all?) Chinese words come in "short" and "long" forms. Native speakers switch back and forth between these forms effortlessly as the situation requires. In many cases, fixed expressions can take one form but not the other. Just another thing we students of Chinese have to learn. :)

By the way, there is one "grammatical" difference between 学 and 学习 that I can think of off the top of my head. 学习 can be used as a noun to mean "studies" or "academic skill", e.g.

我学习很好. -> "I am good in my studies." or "I do well in school."

(I'm not sure if this is technically a noun. I think most people would gloss it as a nested topic-comment structure: 我//学习//很好.)

P.S. Most of my knowledge about this topic comes from the excellent "Chinese: A Comprehensive Grammar" published by Routledge.