Hanging out vs hang out

English as a Second Language English Vocabulary, Grammar and Idioms
1
  1. We do hanging out with our guest. ( I unmistakenly heard this construction from someone) and I thought it would have been the following:
  2. we do hang out with our guest. Please help me which one goes.
2

“We do hanging out with our guest” is incorrect. This grammatical pattern is possible, for example in “I do painting with my friend”, but it would not be used with “hanging out”.

“We do hang out with our guest” is possible, but for me, “hang out with” (very casual, usually used in relation to friends) is not an obvious fit with “our guest” (more formal). “do” adds an emphasis that would only be required in certain contexts.

3

As Dozy explains, this leaves:
We hang out with our guests.
We are hanging out with our guests.

4
  1. I do painting with my friend.
  2. I do paint with with my friend.
    Please reconfirm if the examples could be used interchangeably. But thanks for your elaboration.
5

(2) is not really possible, except as an emphasis in a conversation similar to:
I paint with my friend.
I don’t believe it.
It’s true, I do paint with my friend.

Although ‘I do painting with…’ is acceptable, ‘I paint with…’ sounds better in most cases.