No, "homework" does not need a capital letter unless it is at the beginning of a sentence or used in a title.
No, "thrush" does not need a capital letter unless it is at the beginning of a sentence or part of a proper noun.
NO
no
not really
No
The word king does not need a capital letter. The reason the words April, California, and Renae need capital letters at the beginning is that they are proper nouns.
s
YES
farm
No
No, there is no need for a capital letter after an equal sign in most cases. The convention is to start a new sentence with a capital letter and not necessarily after an equal sign within a sentence.