Here's another (unlikely) possibility - though I will say that I think the advice of "run now" is more likely a possibility: he may have financial and/or past contractual (marriage) problems which prohibit him from purchasing a gun, and he may not have been forthright with you about his personal information for that very reason. He may not have wanted that to come forward if it's still fairly early in the relationship. Hypothetical: maybe his vengeful ex-wife has a restraining order on him, and he's had to "fly under the radar" to avoid the repercussions of it, or maybe he's got someone claiming he's her baby's daddy, and somehow he got into a child support DB and is restricted from purchasing a gun through that (which I believe is possible, at least in some states, but I have no personal experience with any of this).
Also, if you ran background checks on him, how were you able to find out his proper age without finding out if he had a felony record and various other related information? IIRC that information is generally in the background check. Did you do the background check on his SSN, or some other information such as name, and possibly got the wrong person (is his name common?)?
I'd say confront him with his lies, and ask him why he's been doing it. Maybe he told you lies initially thinking it'd never amount to anything (in terms of a relationship), and now it has, and he's stuck between a rock and a hard place trying to decide whether to tell you or not - on one hand, he'd have been lying to you which would anger you, and on the other hand it's going well for the time being (in his mind), so the trade off is probably fairly severe.
Now, you may not want to have anything to do with someone who lies, but I'm just trying to interject the very slim possibility that there may be legitimate, morally upright reasons for the lies.
AGAIN, let me say that it sounds like a situation where you would be best to RUN AWAY, but without actually seeing and meeting the guy (I'm a very good read on character), I'd be hard pressed to make an assessment based solely on the information provided thus far. I'd say get your affairs in order (and your material possessions out of his grasp, maybe change the locks to your place if you suspect he may have a key, etc.) and then confront him with these things in a public place while carrying (coffee shop, maybe).