A) The telephone is ringing. I'll go and answer it.
"And" is often omitted by native speakers, so your impression was right there.
B) I feel like going dancing tonight.
Sounds nice and natural to me, since we refer to this activity as "dancing". I can see how you might be worried about having two "-ing" verbs adjacent, but it's honestly a perfect construction. Your version: "I feel like going to dance", is also perfectly valid, but I just feel that "going dancing" sounds more natural.
EDIT: (See @Tristan's comment)
The British are unlikely to omit "and", as in "and answer the phone". The Americans are more likely to omit it.