The first is an example of "headline English", about which you will find many questions on this site. Headline English omits as many unnecessary words as possible, leaving only the bare meaning. In some cases you have to understand the context to get the meaning, but in this sentence, imagine it was written as a complete sentence
[The] Sheriff's hiring of [his] political supporters [has come] under fire
Then, figure out the subject of the sentence, which in this case is the gerund "hiring". "Sheriff's" is an adjective saying whose hiring it was, and "of his political supporters" is just a phrase telling us who was hired. So it's the act of hiring that is the problem.
'Mystery meat' servings in school cafeteria proclaimed 'inedible' by students.
Again, what exactly have the students proclaimed "inedible"? The school, the cafeteria, the servings, the meat, the mystery? Obviously not some of these, but, again, think of this as a complete sentence
Servings (of 'mystery meat') (in the school cafeteria) have been called "inedible" (by students).
This should make it clear that the subject is "servings of mystery meat".
Your second question should probably be asked in a separate post, but the short answer is that "that/which" is often optional. It's likely many English learners are told it is required, so they become familiar with the structure, but once you read more you'll find that native speakers take all kinds of shortcuts. Examples:
Please bring me the book (that) you bought for me last Christmas.
Sally goes to the same school (that) her mother went to as a girl.