Why does she say "that's right" here?

Is “lay” the past tense of “lie”? Or “lay” is “lay” (present) here? Could you explain a bit?

"To lay" and "to lie" are different verbs with similar meanings. Lay means "to place something down." It is something you do to something else and is a transitive verb. Lie means "to recline" or "be placed." It does not act on anything or anyone else and is an intransitive verb. The past tense of "to lay" is laid, while the past tense of "to lie" is "lay". Ex. I laid down the book on this table yesterday, but now I can't seem to find it anywhere. Ex. I lay awake in bed all night, unable to sleep.

Can I also use the word "there" when I want to say farewell to someone?

No, the word "there" is not used when saying farewell or goodbye. The word "there" is used as a way to say: "I see you there" or "there you are!". A kind of acknowledgment of someone's presence. You could, however, use "then" when saying goodbye as a way of moving on and going to do the next thing. It is quite common. Ex: It was great chatting with you! Goodbye then. Ex: Hello there! How are you doing?

What is y’all?

Y'all is a contraction of you and all. Ex: Let me sing it for y'all!

What does "embark on" mean?

To "embark" on something means to start or begin something especially exciting, challenging, or important. This word is often used in the context of traveling or beginning a long journey. Ex: Tomorrow we are embarking on a road trip across the country. Ex: They moved and embarked on a new life in Australia.

What does "first of all" mean?

"First of all" here is used to introduce the first point someone is making. It can also mean "before you do anything else", and is said when you want to say or ask something before a conversation proceeds. Ex: First of all, I'm reliable, second of all, I'm a hard worker, therefore you should hire me. Ex: First of all, please tell me your name.