Is there a more sophisticated expression that indicates firing someone from the position?

Is it okay to skip the noun after superlative? Shouldn't he say "one of my brightest and best STUDENT"?

Here, "students" is implied and understood. Therefore, it is okay to not include "students" here.

What's "all along" mean and when's it used?

"All along" means "the whole time" or from the beginning of something. It's used when you want to mention the whole duration of something while something else happens. It's also often used when something happens without knowing or being aware of it until a point in time, like a contrast of two things. Ex: You were lying to me all along?! Ex: Maybe something was wrong all along. Ex: You were honest all along, but I didn't believe you. = You were honest this whole time, but I didn't believe you.

I heard that illegal aliens sometimes get deported outside the border. And let's assume their newborn babies automatically got citizenship. Then, what happens to those children if their parents are deported by authorities?

From what I know, in the US, the parents could be deported, but the children will probably be taken care of by the country's childcare services. They become a "ward of the state," placed under the care of a guardian or legal entity. In Ontario, children have the right to go to school and have an education, even if their parents are illegal immigrants.

Can I say "last one thing" instead of "one last thing"?

The expression itself is "one last thing." "Last one thing" has the wrong word order and would sound awkward if used. Ex: One last thing! Can you take out the trash after doing the dishes? Ex: I have one last thing to discuss with you.

What is the meaning of "take someone up in one's head" in this situation?

To "take them someone up in one's head" in this context just means to think about a person or keep them on your mind and have compassion towards them. This is not a general expression and is not used commonly in other situations.