Why "underneath" is used here? why not just saying "under"?

Is the expression "what do you say?" used for asking the other person's opinion? How do you use it in general?

You are correct. "What do you say" is an informal expression used when asking for another's opinion, particularly when you want to convince someone. The answer is often a yes or no answer. Ex: A: Dinner on Saturday. What do you say? B: I'm there! Ex: A: I think Joey should be the project leader, based on seniority. What do you say? B: Sounds good to me.

What does "lab" mean?

Here "lab" is a short form of the noun "laboratory," which is a room or building equipped for science experiments, research, training, or manufacturing drugs or chemicals. Ex: My school has a really nice biology lab. Ex: The drug will go through a series of lab tests before it's released to the public.

Is "Could I please ~" a common expression?

Yes, "could I please" is common and is a more polite way to say "can I please". "Could" can be used in replacement of "can" to sound more polite when making requests. This is because "could" is a conditional verb and is used to express possibility instead of certainty like "can". Ex: Could I have a glass of water?

What does "a sense of" mean?

Very good question. When thinking about the meaning of "sense of" there is a difference in meaning between "sense" and "sensation". It is true that sense can mean feeling, like a "sense of tiredness" or a "sense of hunger". However it can also, when in the context of an ideal or a virtue or an ability (such as "humor" or "justice" or "irony" or "direction" or "responsibility"), mean the ability to appreciate or understand something. This is how it is being used here. So she is saying that the customers in these restaurants feel or understand the idea of urgency, of not having a lot of time. Ex: My husband has a great sense of humour. Ex: I have a terrible sense of direction. I'm always getting lost.

Why "underneath" is used here? why not just saying "under"?

"Under" is used when something is directly under another thing. "Underneath" is often used to say that something is covered by another thing. Ex: My dog used to sleep under our bed, but he's grown too big now to do that. Ex: This jacket's too small for me to wear a jumper underneath.