I just bought a brand new Giro Shiv helmet. However, it was in my checked luggage on my flight, and somehow it got a few dents in the side. Does this make the helmet substantially less safe and do I have to replace my brand new helmet?
Your helmet has been compromised. To what extent, no one can tell you. Basically, a helmet is rated for 1 impact when it is in mint condition. Airline travel, kicking your helmet, tossing it in your car... all those things will significantly reduce the lifespan of your helmet. And you do not know how it will affect the safety of your helmet until it is too late and you end up with a concussion.
Best and safest bet, is too replace it. Next time, take it with you in carry-on so you can keep an eye on it.