Perhaps you could use their words to your benefit.

Misdemeanors and felonies differ, in definition, from state to state. What is a crime in one state may not be in another.
Laws change from year to year. What is a crime one year may not be the next, or vice versa.

Until you are tried and convicted, your activities are only your "alleged crimes" , thus unless convicted you have no other misdemeanors and felonies than those on record.

By the way, if you were convicted of a felony, they should not have let you into the academy if they knew you would have access to firearms.

MY BIG QUESTION": What is the statute of limitations on being able to convict you on any of the things to which you admit ?

I would be concerned with the fact that that may keep your statement of admission of guilt to specific things as a record.
EX: I stole some video games from the Walmart on 120th/Sheridan when I was 14.

You never know at what time in your life that list may 'go public'...even if it is never supposed to.

If you choose to admit anything, I'd be general and vague: EX: Shoplifted some things when I was a kid vs. the above Walmart example.

I might even spend a few hundred bucks for a good Criminal and/or Employment Lawyer and get some advice about what they CAN and CAN'T require you to do, under the law.
I was told by a lawyer, when a company wanted me to authorize them to look at any of my personal financial records as a condition to hire me, that 'They can do this...If you want the job, then sign it, else don't sign it"

Somehow I kind of feel like it is the police trying to get around the law. Similar to how they often tell you "Whatever you say can and will be used against you." but then try to use tactics to get you to talk ("We'll go easy on you if you cooperate", "Mind if we step in your house for a minute to talk ?" etc.)

I wish you the best of luck in your decision.

It is not an easy thing to freely admit guilt...just ask any Catholic that has gone or still goes to Confession.