In your example, no it shouldn't be capitalized. Only capitalize it when you're using the name of the institution as a proper noun. Whether you're referring to a specific one isn't what's relevant, but rather whether you're referring to it by name.

Yes or no. You haven’t given enough context to decide. If you use “University” as a shortened name for a particular university, say the University of Nebraska, then yes, it should be capitalized.

Yes, when you are referring to a specific university, it is customary to capitalize the word "University." For example, "The University is ready to help you." This capitalization indicates that you are talking about a particular and distinct institution. However, when you are referring to universities in a general sense, without specifying a particular one, you would use lowercase, as in "Many universities offer online courses."