No Stupid Questions

There are specific visas for athletes that allow them to bring in regulation meeting sporting equipment.

Getting a firearm across country borders is not as hard as people think as long as you research the requirements of the countries you transit plus the airlines beforehand.
Some countries will have conflicting laws though.
From memory importing a firearm out of the UK requires the case to be stickered as contains a firearm where as US laws are the case mustn't show it contains firearms. (My info on this may be out of date)

Oh! I actually can personally answer this one. I’ve been a competitive air gun and rimfire .22 LR for years and even attended college on a rifle program scholarship at a D1 program (yes! The US has a major collegiate rifle program)

Traveling internationally depended much on where you were going. We competed at the international junior Olympic championships just outside of London one year in the UK. We had to package all ammo and firearms separately, in locked containers which ultimately wasn’t that big a deal. As with flying with firearms in the US, it ultimately comes down to how familiar those such as gate agents and check-in staff are at the airport with their country’s firearm travel laws. Flying out of the US was always easy as they’re used to firearms, but flying back to the US was sometimes a nightmare and we’d get to the airport up to 6 hours before a flight to deal with any BS.

The simple answer? You declare it at customs and make sure to follow the relevant laws.

Any questions you might have can be answered by the appropriate embassy/state department. It’s part of the job, after all.

One thing to note. Always, always clear out your bags- everything you’ll be taking and pack them carefully. Make sure not to have anything that would be illegal to where you’re going; even if it’s legal where you are. (And this includes countries you’ll pass through.)

You will be searched and more than likely it will be found. You don’t want to be the next Britney Griner only to find out that you’re not enough of a superstar to illicit media interest.

Idk about internationally, but you can check a firearm on US domestic flights. You just declare it, and check it in as baggage. These guns for the Olympics are air pistols, so the policies are probably pretty lenient in most countries.

Hunters fly with firearms sometimes. It just depends on the customs regulations. I've seen it at the airport. You check the bag and declare it. I'm sure there's some amount of paperwork, depending on the country.