What you just described is straight-out fraud. And if you send that application through the mail, it is a federal crime. Look at it this way -- you're lying to people to get a benefit from them, and whether it is money or a slot in a school doesn't make a legal difference. It's no different from some fraud through mail scam where you get people to send money to you by lying about being sick, etc..
If I submit an knowingly inaccurate application to a school, is that a crime?
Technically, yes. Although it is unlikely to be prosecuted in most cases.
If I pay an individual in a school to get me into the school, has the individual committed a crime if they help me? Has the school committed a crime?
Depends on whether or not fraud is committed at some point, and whether or not you know the person you paid is committing fraud.
“What you just described is straight-out fraud”
Thanks for the informative post. It makes sense that if I lie to get a benefit, that should be a crime.