Not sure if this answers your question,  but I usually say to friends, "Hey, wanna grab a coffee?"

To a co-worker or in a more formal setting- " Would you like grab a coffee after lunch/ later on/ tomorrow morning?"

Slang for coffee in America is "a cup of joe" pronounced "cup-pa-joe". Older people may use this slang. "Do you want a cup of joe?"

I live in California so the phrase can be region & age based.

Yes, you do "ask someone" or "invite them to" coffee. "Did you ask her to coffee?"/ "Did you invite her to coffee on Thursday?" 

English question

--

Usually, I say--"Hey, want to grab a coffee?" or "Would you like to grab a coffee?"

In the U.S, inviting someone to coffee often can be implied as a date (at least among young adults, haha)--unless you're already friends or you are inviting someone to coffee for a more specific reason. Like, "let's grab coffee before work" or "Would you like to grab coffee first?"

English question

Free to have coffee with me?