If you are talking casually about a school that has a longer formal name, then no, you would not capitalize. It's possible that there is a club or restaurant somewhere named "The University", in which case you would capitalize.
No, except at the beginning of a sentence because it is n ot a proper n ou n.
Yes, becauseit is a proper n ou n. Example: Lourdes School of Makati Harvard University
Vanderbilt University, all of the Ivy League Schools and there are many, many more!
use I or a name of a town
Yes. 'Sun' is its name and you capitalize it just as you capitalize anyone's name.
no, university would be exactly as the word school or chair or table. eg: what university will he attend? as to : he is planning to go to the University of Manitoba.
If its a name.
Yes. It is Jane Doe, M.A., Ph.D. (Note the lower case h in Ph.D.)
You capitalize Apollo because it is a name.
No, but you capitalize the name of a biome.
Yes, you should capitalize the name of a feasibility study as it is a proper noun.