What is the difference in meaning between the verbs “think”, “feel”, and “believe”?

Contrary to what many writers seem to think, these three words cannot be used interchangeably!

Think is used for expressing an opinion: Tom thinks that his best customer will renew the current supply contract.

Feel is used with regard to the sense of touch or a person’s health or emotional state: She feels good today. And: I feel uncomfortable about the deal.

Believe is much stronger than think and refers to a person’s deeper convictions: Mr Wong believes that now is the right time to expand. 

Can you explain the meaning of: "He said he would stay with her until she got the hang of operating the machine"?

The idiom to get the hang of something means to understand how to do something. In your example the man is saying he will stay with the woman until she understands how the machine works. Now, if you are having trouble understanding this I’ll offer various explanations until you get the hang of what I’m saying.

Could you explain the meaning of the idiom "up in the air"?

"Up in the air" is a common idiom which means uncertain or undecided. If someone has no clear opinion on what to do about something, they are "up in the air " about it. For example: Paul is up in the air about whether to ask Jennifer out on a date.

The idiom can also be used to describe uncertain situations: The whole future of the project is up in the air. This means that it is uncertain whether or not the project will go ahead or be continued.