You need a carry permit, but if you don’t have one, you just need to transport your gun disassembled in Indiana. The advice was “and in a box, in your trunk.” Or, at least back when I bought one for target shooting (that I don’t have anymore because I have a child in the house), that was the law.
Some gun dealers who have ranges will sell you a gun and rent you a locker, so you can keep your gun there and come to the range, get your gun, shoot, and put it back in the locker. The ones I saw were behind the counter, and you had to show ID, and an employee got your gun for you.
In Indiana, when I got a carry permit, I was asked whether I’d ever been under the care of a psychiatrist, or been diagnosed with a mental illness. I was told that it wasn’t an automatic denial, but that you needed a letter from your shrink saying you weren’t a danger to yourself or others, and it had to be signed off by the sheriff or police chief, or someone (I don’t remember the details-- I didn’t need this, so I didn’t pay that much attention).
If someone has lied on this form to obtain a carry permit, they have committed fraud, which is some kind of crime, but I don’t know how serious. However, someone actually carrying with a fraudulently obtained permit is committing a more serious offense. I’m pretty sure that, at a minimum, the gun can be confiscated. If you know someone who has lied, you can report them, but I don’t know how much of a priority it will be for the police. We’ve actually had a rash of shootings in Indianapolis, and several are unsolved, so maybe it would be taken seriously. I have no idea.
How any of this applies to hunting, weapons used for hunting, and hunting licenses, I don’t know. It’s just for handguns.