I know both mean "to study" and I also know that while 学 (xué) is transitive, 学习 (xué xí) can be both intransitive and transitive, for example:
But are there other differences? Also in usage and meaning? My grammar textbook was helpful enough but I'd like to hear something from the native speakers too. :)
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.
I don't know the answer from a grammatical point of view, but I know both character's etymology which can give you a hint.
学 in its traditional form is 學, an ideogram of two hands putting something (knowledge?) into a child's head, thus teaching, and, from the child's standpoint, learning.
习 in its traditional form is 習, two feathered wings on top of white (originally a sun) and it meant learning or practicing how to fly, thus, practice.
So while 学 means basically to learn, 习 moves the meaning more towards learning by practice, but this is pure speculation.
When you learn Chinese, you will get more if you know the origin of the words.
'学习' origins form the ancient sentence '学而时习之', which means that you learn (学) theory and use (习) the theory correctly. So the 学 and '学习' are a little different although you can exchange them in most cases.
I'd also add that 学 by itself feels incomplete. In most sentences, 2 characters will be required for clarity, meaning and balance. For example:
学着 (xue2zhe) studying <- I am not 100% certain if this is acceptable. Native speakers care to comment?
学会 (xue2hui4) here 会 is the complement of 学 and means that after studying, one is able to do something
学到手 (xue2dao4shou3) is also a completed form of the verb, it means to have obtained some knowledge or skill through study
学完 (xue2wan2) to finish studying
学好 (xue2hao3) either to finish studying successfully or to study well (i.e. to learn from good examples)
学 is gaining knowledge or skill. 习 is to do practice. 学习 is to study something, it can be converted to a noun to mean the behavior of studying.
「学」 can also mean "to imitate" or "to repeat after <somebody>", whereas 「学习」 does not have this meaning.
1和3意思一样，学和学习在不同的地方表达的含义不同，但是上面的1和3句子中，差不多是一样的。
我个人觉得，学习，一般是指学习知识比较多。学，就包含更多了，例如学做饭，一般不太会说学习做饭。
I'm still a learner but I think 学 is study in general and 学习 is doing exercises, learning something
