"Speak" and "speak up", how are those different?

What does "mental that one" mean? What kind of expression is it?

This is a British expression that means "crazy" or "out of their mind." I am not British, but I believe this expression is slang. When Ron says, "Mental that one," he tells Harry that he thinks Hermione is crazy. Ex: She's completely mental. Ex: Going shopping before a holiday is mental.

Does "I don't think you deserve to be there." mean "You don't have qualification to be president"?

Yes, you are correct. To not "deserve" to be somewhere/something means that you are not good enough or qualified enough to be holding that position. The speaker is stating her opinion that Trump does not deserve to be President, and that he needs to resign. Ex: You don't deserve that position. You're not even qualified. Ex: You deserve to be there. You have earned it.

Why is it "inspiration for" instead of "inspiration of"? Can I use "inspiration of" here?

"Inspiration for" is correct here. "Inspiration for" is used here because this sentence is saying that Elon Musk "gave" inspiration "for" Robert Downey Jr.'s version of Tony Stark. Therefore, "inspiration of" is not correct to use here, and is rarely used in general. Ex: Roses are the inspiration for my painting.

What is "Echinacea"? Is it a drug?

"Echinacea" is a supplement/herbal remedy made from flowers that supposedly helps with inflammation, treating colds, boosting one's immune system, etc. In this show, the main character has cancer and his wife has him take Echinacea to try to keep him healthy.

"Speak" and "speak up", how are those different?

"Speak up" is a phrasal verb of "speak". To "speak up" in this context means to voice one's opinion or say what's on one's mind. In other contexts, "speak up" can mean to speak more loudly. Ex: If you need help, speak up. Ex: It is time for us to speak up for our rights.