Soon, I am traveling from Kolkata to San Francisco with a 20-day stopover in Abu Dhabi. I'm an Indian and a UAE residence visa holder, so the long stopover at Abu Dhabi isn't an issue. I'm planning to get my US visa stamped at Abu Dhabi in those 20 days I'm going to get. Will the airline staff allow me to board my flight at Calcutta just for the Abu Dhabi leg flight without the US visa stamped on my passport?
Since your US flight is 20 days ahead, your luggage won't be checked through, and you won't be issued a boarding pass for the second flight yet, airline employees will let you through.
In general airlines only care about not having to transport you back (and paying a hefty fine) in case you land into a country without a visa. Since you're a UAE residence permit holder this obviously doesn't apply.