As a relative pronoun, what's the main difference between "that" and "which?" How do I distinguish the role of these words?

Is there any difference in nuance between "doing that dancing" and "doing that dance"?

The difference would be that usually a "dance" would refer to a specific dance. For example the salsa or flamenco. Whereas the word "dancing" means the general activity. Since he is not doing a specific type of dance, he's kind of just moving around, the interviewer chose to say "that dancing". However many people also use dance non-specifically so, depending on where you are, "that dance" would also be accepted in this sentence structure. This is one of those differences that are affected by context and region. Ex: Do you want to go dancing? Ex: Do you want to go dance? In the UK both if the above examples would be acceptable forms, the second being more informal.

What's "prim" mean?

"Prim" means severely formal and respectable. Here it's almost like calling her "uptight" and like she can't relax. Ex: You don't have to be so prim about planning, have some fun too! Ex: I've never considered myself a prim young lady despite my mother's wishes.

What does "measure" mean here?

In this the word "measure" is used to mean a way of achieving something, or a method for dealing with a situation. You can think of it as being similar to "steps" or "methods". Ex: What further measures can we take to avoid losing business? Ex: These measures will help us stay safe. Ex: We have safety measures in case of an emergency.

When do people say "remind me?" I hear people say it often.

Yes, you will often hear people use the informal expression "remind me to X" to ask to be reminded of, prompted to, or be told to do/say something. Ex: Remind me to water the plants later. (The person is afraid they will forget, so they are being asked to be reminded later.) Ex: Can you remind me to do the dishes later? Ex: Don't forget to remind me to tell you my funny story.

For an exaggeration, what phrases can I use in order to replace "really" in this context?

To replace "really" here, you can use synonyms like "very," "extremely," or "quite." Ex: George was very hungry. Ex: The weather is quite sunny today.