Caregiver is correct, just having a cut will NOT cause an air embolism.
The answer is no. You would need entry to a blood vessel. A "cut" would not ordinarily go that deeply. If vessels are entirely cut they retract. I have seen many people with severed arms at accident scenes, and they have not suffered an embolism and survived.
Yes - IV lines can get air into your blood, but sometimes the blood in the vena cava is below atmospheric pressure (due to the heart sucking the blood from that vessel into the heart) and it is possible to get air sucked into your blood by that route, but it is nearly impossible, since if you got a cut anywhere on that blood vessel, you'd probably be dead anyway (and surgeons have developed methods to get around this fact with heart surgery and that kind of thing). it has to be injected into the blood vessel in some way with an IV drip or compressed air blast to the skin
The only way to get an air embolism is from IV lines with too much air in them or a direct injection of air in excess of 3 cc's.
No, there would have to be a stronger force than just the regular atmospheric pressure to allow the air to get inside the body.