Learn English – The opposite word of “to meet up with someone.”

Not in the way you are using it.

"Small is the opposite of big" is saying that the two words mean opposing things. This is okay, because "of" indicates a difference in essence.

"Small is opposite to big", however, indicates an opposition in physical position. It suggests that "X" is in the opposite position compared to "Y". The "to" in this phrase indicates a physical opposition, where the words "small" and "big" do not have physical entities to represent them, and it would only make sense if you were putting the words physically in a list, and saying they were opposite to each other on that list.

British slang has lots of phrases for this, including 'fancy woman' and 'bit on the side', but I don't know if they'd be understood outside Britain.

'Paramour' is an archaic word. I wouldn't use it at all unless I was writing a fantasy novel. The same goes for 'concubine'.

'Mistress' doesn't have the connotations you describe to me, and is the word I would use if it wasn't already clear that the man was married. If it was, I would probably say 'girlfriend'.

A term that could be used for either gender is 'lover'.