1.

I have got to go now.   - Typically you'd say "I've got to go now" instead or even "I gotta go now". It's a more casual way of saying "I have to go now". It's a way of finishing phone conversation for example.

"I have to" vs "I need to" is a bit difficult to explain. They both imply a similar thing - they carry more importance than "I want to".

"Have" can often be about external circumstances, things that are outside your control.

=> "I have to study because I need good grades."  - So in order to get what you "need" there are certain things you "have" to do. 

"Need" on the other hand is more about your personal desires. 

=> "I need to be more organised." - It's my goal to be better organised because it's helpful for my goals.

So going back to your question. Saying "I need to go now" is less final and perhaps a bit more polite than "I have to go now". "Have to" implies you really have less choice than "need to". But both are perfectly correct in this case and can be used interchangeably.  There's a subtle difference in meaning.

2.
I was able to dance with her - This means you had the chance to dance with her and you did.

It's also about the fact that you were able to. So perhaps it was a difficult dance or there was something that made it more challenging, so you're stressing the importance of the fact you were able to do it.

=> I was able to pass the test. vs I passed the test. 

=> I was able to prove them wrong! vs I proved them wrong.

I could dance with her.

This means you have the chance to dance with her, but doesn't mean that you did. You just have the option. It's also in the present tense. It almost sounds like "Well, I could dance with her, but I might not." It's an odd one to say when it's about dancing with another person because it implies the speaker is the only one who can decide, and obviously dancing takes two people. 

I hope this makes sense :)

What is the difference between two grammatical forms below?