Wow that sucks! I think a lot of us have to check yes, but a lot of us can explain away an incident like drunk & disorderly with a "boys will be boys" thing. Solicitation not so much
I would suggest contacting an attorney in that jurisdiction to see what can be done. I realize $ may be tight now but a respectable attorney will do a free consult & only charge you if she actually does something.
As for what actually will show up on a background check will vary depending on what kind of a check the company invests in: the company pays for them and a thorough one isn't cheap. Amazing how many firms are penny wise pound foolish.
That said, even if a company only asks about felonies on its app, be advised that there is a good chance your mistake may come back on the report anyway. It is typical to sign a waiver authorizing them to do a complete check & if they do everything will come back.
The good news is in most cases up to senior levels you will have a verbal offer before they ever order a check (again these things cost $ & they don't waste it on every applicant). At that point hopefully at least the hiring manager likes you & a heart felt "I was really just in the wrong place at the wrong time" routine & see how it plays.
IANAL, but a word about lying on an application. In most cases it is grounds for termination down the road, but it isn't a crime or at least a crime that would be prosecuted. On the otherhand, if it is anything to do with the government (including things like applications to state licensing boards) lies are often considered perjury and can get you back into deep shit criminally.