Does the meaning get changed if I replace "whatever" with "what"?

What does "get down to cases" mean?

"Get down to cases" is an idiom that means to begin being serious and doing something that needs to be done. I have never heard this idiom before! It's more common to say, "let's get down to business." However, since this cartoon is older, this may have been a popular idiom at the time. Ex: Let's get down to cases. We have a lot of work ahead of us. Ex: We need to get down to cases if we are going to finish on time.

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

There are a few expressions that you could use. One would be "let go," "terminated," or "cut." The nicest way to say "fired" would be "let go." Ex: I'm sorry, Jim, but we have to let you go. Ex: My position was terminated yesterday. Ex: Her job got cut because of the company merge.

Is "kick to the curb" an idiom? What does it mean?

Yes you are correct. "Kick for the curb" is an idiom meaning to discard, abandon, or dismiss someone or something that has become redundant, obsolete, useless, or unwanted. Ex: I can't believe that after 20 years the company would just kick me to the curb like that! Ex: I think it's about time we kick this old computer to the curb. Ex: I heard Jenny kicked her boyfriend to the curb last night. They must have had an awful fight!

Is it common to use "buttoned up" like this? What is the meaning of it?

"Buttoned up" in this context is a metaphor for making everything secure, closed, and ready to go. This expression can be used in other contexts to mean to close, secure, or finish something, although it is not widely common. Ex: Let me button up my assignment and then I'll come along.

What does the "Pipe dream" mean?

A 'Pipe dream' is an unrealistic hope or fantasy. The phrase 'Pipe dream' is an allusion to the dreams experienced by smokers of opium pipes.