I would like to find a job in a (no 'a'??
No. You need "a".) university/school/church/camp. -- if I am thinking in terms of a distinct university, not necessarily the building, can I use 'a'?
No. eg, A university next to my store is one with rich history.
Sounds very strange. You need The university ... Otherwise, it sounds as if you are saying "One of the (possibly many) universities next to my store ..."
#1 line of hospital tapes -- I think the word 'tape' is uncountable, not variable, and it would seem unlikely to be made a plural without a good adjective or context.
The change to the plural tapes here indicates "kinds of tape". A "line" is a series, a variety of different kinds of tape.The types of jobs favored by people living here are professor, computer programmer, and driver. -- no article OK before the words professor, computer programmer, and driver? Right. No article is good.
Typical definition of a word: An *** is a small box with a microphone which is connected to a ... -- what difference would it make if written as "An *** is the small box with a microphone which is connected to a ... It would not make much difference. Sometimes a change of article doesn't make a lot of difference. Nevertheless, it would sound as if you were saying "An *** is the small box -- you know which box I'm talking about -- with a microphone which is ...". And that would seem a little strange to most speakers of English. It makes the definition lose its generality, which is what we normally want in a definition.
CJ