If I can avoid it yes but they have on occasion...
Why are they crying? If they are hurt, I comfort them, if they are throwing a fit, I put them in timeout until they are ready to behave.
Nope. We leave. Other people don't need to hear that.
I will try to quiet my kid down first. If the fit/crying doesn't stop after a few minutes, I will drag them out the door and we'll go home with nothing. I have had to do it once or twice.