Wonder if the DC shooter was upset that his pay didn't arrive on time? Seems 40,000 federal workers who had direct deposit weren't paid on Friday due to a computer glitch. http://www.standard.net/stories/2013/09/17/pay-delayed-40000-federal-workers. Either that or one of his friends didn't get paid. At any rate it's a given that he had paranoid feelings against his coworkers. I've seen that develop in the workplace before. A worker thinking that people are plotting against them.
Now a little background on security and the checks involved. All require reporting of any felony convictions. But it depends on the degree of sensitivity and the area of access. For some government buildings you can't have so much as an outstanding parking ticket and expect access. I know because I've worked at most facilities here. The problem is that much of government work is contracted. Many in the trades do have criminal records but that doesn't necessarily disqualify them. If it did we'd have a hard time getting anyone to get many of the construction jobs completed.
Last I checked firing a weapon in city limits is a finable offense. The number of police reports on out of control people would surprise you, so common now that many aren't even reported. Hearing voices in your head is a mental issue not a police matter and that's they way they look at it. Unless he was armed and threatening at the time the police would just leave.
Two other points in this case are that the shooter had already received security clearance. Once in the system an individual can have access to where ever they are permitted. And finally with the cut backs and everyone wanting to save money background checks are now outsourced to other contractors.