Most departments will not hire someone with a criminal record currently, but it will depend on the type of crime committed and the circumstances surrounding it. Violent crime, no. Any kind of domestic violence, no because you legally cannot carry a firearm. Misdemeanor property crime, maybe. Drugs, anything but marijuana no, and maybe not even then depending on how hardline they want to be.
As far as mental health, there is a lot of negative stigma in the United States about mental illness. There's even more when you bring gun control into the conversation, as most people agree that access to firearms should be restricted for people with mental illness, due to the potentially increased risk of violence against themselves or others. I'm not saying that is or isn't accurate, as I haven't researched it, but that is the general feeling put out there in gun control conversations, which law enforcement has a vested interest in.
It depends on a lot of factors honestly, the most important one being the applicant pool itself. Most police departments have a very limited number of openings every year, and as a result get to be very selective about who they hire.
When you have 1000 people who apply for maybe 10 positions that you have available (which is pretty accurate for an average sized department in the US on an annual basis), you can make a lot of easy cuts via minimum qualifications and testing. That's why there are minimum education requirements, physical fitness tests, psychological testing, polygraph tests, etc. It's very easy to make a first round cut and say anybody that doesn't have an Associate's degree or higher, or has a criminal record does not get considered. It's nothing against the applicant as a person, it's all numbers at that point.