Translation:He invited me to dance last night.
"He invited me to go dancing last night." should be accepted. Otherwise, it should be "He asked me to dance last night."
"He invited me to dance last night." is unnatural.
I think "He invited me to dance last night." is natural and makes sense. In fact, I use it a lot to talk to my family in Chinese and English.
Yes. Unless that he is a showbiz owner and the speaker is a dance performer.
"He invited me out dancing last night" didn't work, but I think that it should.
"He invited me to dance" is not a good English sentence--it's not the way it would usually be said. In fact its meaning is unclear: did he see me at the bar and ask me to dance with him? did he see me sitting down and encourage me to get up and dance with everyone? did he ask me on a date to go out dancing?
It maybe a bit "formal" and as was stated "asked" would be the more common phrasing in English, but it is good English and is unclear because without context the Chinese is also "unclear" with a broad range of possible specific meanings. That's just how Duo works.
Is 请 really used as a verb like this in daily Chinese or this this very formal language like when using 贵 to mean "your" as in 贵司 and 贵国?