No, it is not a proper noun. It is a common noun and does not need to have a cap.
No, "homework" does not need a capital letter unless it is at the beginning of a sentence or used in a title.
The state capital of Montana is Helena. Glacier National Park is in Montana.
No, "thrush" does not need a capital letter unless it is at the beginning of a sentence or part of a proper noun.
No.
Helena is the state capital in Montana.
NO
no
not really
No
The abbreviation of units is usually written with a capital letter, if it is based on the name of a person.
The state capital of Montana is Helena.
Helena is the state capital in the U.S. state of Montana.