For example, can I refer to the main gate of a university as the school's main gate? Or say school begins in September instead of university begins in September (especially in informal speech)?
Merriam-Webster defines university as:
a school that offers courses leading to a degree (such as a bachelor's, master's, or doctoral degree) and where research is done
Therefore a university is a type of school and you can use the word school to refer to a university.
Here is an example of Vanderbilt University referring to their "school's main gate":
More than 1,800 new Vanderbilt University students, including about 1,600 first-year students and more than 200 transfer students, will march into the school’s main gate Sunday, Aug. 23, at 5 p.m. during Founders Walk, a tradition that formally welcomes new students to the Vanderbilt community.
It is also perfectly acceptable to say "school begins in September", although some people might prefer to say "classes begin in September".
Note: As mentioned in the comments, this answer only applies to American English.
In American English, this usage is correct. In British English (http://www.oxforddictionaries.com/definition/english/school), school is explicitly mentioned as being for children. However, it can also be used to refer to a specific department of a university:
2 Any institution at which instruction is given in a particular discipline: a dancing school
2.1 North American INFORMAL another term for university. Harvard is certainly not a loafer’s school
2.2 department or faculty of a university concerned with a particular subject of study: the School of Medicine
Both of these are educational institutions and they render service for students with different levels of skills. Hence, interchanging the words would not be appropriate.
Another mistake that I see is that you have used 'an' before a university which is incorrect. Please note that 'an' is used before words which start with vowel sounds, not with vowel letters. The word 'university' starts with the sound 'yu.' Hence, you need to use 'a' before a university. Couple of more examples are given below: a unique opportunity a universal truth a useful machine a useless fellow but an umbrella, an unavoidable situation, etc...
Sometimes, I still catch myself to say "I am going to school", even though I am working as a teacher/researcher at the university.