Three separate issues...
The official federal background check (required by TSA for all airport employees) goes back ten years. A specific list of crimes applies, and any of those crimes are disqualifying within ten years. I'm not certain if the airlines get a pass/fail or an actual list of convictions, or how far back it goes. From what I have gathered I suspect they get the full list from the FBI files. Keep in mind that the FBI may not delete records expunged at the state or local levels. For national security purposes, they only care that it happened, not what legal maneuvering came after the fact.
A private background checks is often (usually) done by airlines. This could employ an outside agency, in which case there are legal limitations on what info can be reported...I suspect many airlines go this route. However if an airline wants to do their own background checks (with their own employees, not an agency) they can look at anything they like if they are willing to spend the time and money to dig it up. They may use any of this info in their hiring decision, unless otherwise excluded by law (ie sexual orientation in certain liberal states). This is where they will look at credit reports, lawsuits, and former employers.
The FAA medical thing might be an issue also. An airline could require you to sign a release to obtain a copy of your FAA medical file. You could probably find out in advance when you apply to a specific airline.
Generally, honesty is the best policy, unless you like lying awake at night worrying about what your employer is going to dig up that you forgot to tell them.