Worried Job Offer Will Be Rescinded

Hi all, so I recently got a job offer for a food service position at a local hospital. I am so stoked because my last day at my current job which is working at a local grocery store is this Saturday and I was really worried I was going to be leaving this job unemployed for awhile. However I found out the hospital is using Hireright for my background check and I'm kinda worried. First of all I've heard hireright is a terrible company and a pain in the ass to deal with. But there are two things from my past that I'm worried might get my job offer rescinded.

First and the most serious one about six months ago me and a group of my friends got busted by the police for getting caught doing urban exploration in an abandoned building. None of us were arrested and we just got a trespass violation and had to go to court and pay a fine. I'm not sure if this will show up on some criminal record in the background check.

Second and least serious now I didn't lie on my resume but for one of the past jobs I worked (which was just a summer job as a janitor at a local amusement in between high school graduation and starting freshman year of college) I couldn't remember the month I started working there it was either late June or early July. I honestly can't remember it's been almost five years since then and I just put down June is when I started but I honestly could be wrong and it was July.