1. "Has me/us/them running around..." can be used for anything that keeps you very busy. It can mean walking from place to place, or doing several tasks. In this case, "they" refers to all of the employees at the airport that are involved with making it necessary to go from gate to gate. I use this phrase sometimes. I think it's more common to hear people from the southeast US say it. 2. Yes, I would say "fly by." 3. I think both are correct, but most people would use the first phrase. The second sounds a little more formal. 4. I would say "stabbed" if I were talking about a ball point pen. I would use "stuck" if I were talking about a sewing pin. I think that both are correct, but for some reason I think of "sticking" sewing pins into pin cushions.

You could use the idiom in 1), but it should be plural, "They have us running around like chickenS with THEIR headS cut off"? 2) is good - "flying by" or "flying past" 3) Only "We won't arrive in New York for another 30 minutes" 4) "I stabbed myself with a pen by accident" or "I accidentally stabbed myself with a pen"