>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.