Can I use "Nech sa páči" when I get a compliment?
If not, what should I say instead?
Also, what are all the situations where I can use it?
Ďakujem!

Question

Answer

Not actually. 
The response "Ďakujem." is allright.

"Nech sa páči." is used when you are giving something to someone, or when you did something for someone, he tell you "Ďakujem“ and your response could be " Nech sa páči." It's used when you are passing something or offering something to someone.

Answer

Well, I guess you already understand that the phrase literally means "Let it please you". The most common situation would be when you give someone something.

It really doesn't make sense to use it as a response.  A simple "ďakujem" should be enough.

Answer

"Nech sa páči" as a response to "Ďakujem" is not commonly used in Slovak. The conversation usually looks like this:

Person 1: "Nech sa páči."
Person 2: "Ďakujem."

or e.g.

Person 1: "Nech sa páči."
Person 2: "Ďakujem."
Person 1: "Nemáte za čo." (you're welcome)