What are the benefits of slaughtering an Aqeeqah?

All Perfect Praise be to Allah, The Lord of The Worlds, and may His Peace and Blessings be upon our Prophet Muhammad and upon all of his family and companions.                                                                                                                                                                                                                          Extending thanks to Allah for His grace, expressing happiness for having a newborn, declaring lineage, and feeding the mother to compensate for the blood that she had lost during delivery.

Is a woman`s neck an Awrah (private part)?

All Perfect Praise is due to The Lord of The Worlds.

It is an Awrah, and it is forbidden for her to reveal it before non-Mahrams (Marriageable men). And Allah Knows Best.

Is it permissible to trim the eyebrows if they become excessively long?

Trimming eyebrows is permissible if they become excessively long, yet; it is reprehensible. And Allah Knows Best.

I work in a government institution that gives an additional three dinars per hour I work. The manager combines students into a single session because attendance is low, and the material taught to all groups is the same. I am paid for two hours, although I actually taught only one session, noting that my transportation costs one and a half dinars. What is the ruling?

The only amount you are permitted to take/have is for one hour work. And Allah Knows Best.