No, they do not. However that is the wrong side of an XY problem. You really want your paycheck! As such I will answer that. This is called a "frame challenge" and is legit here.
Also keep in mind they have 14 days to prepare your last check, plus mailing time. It simply may not have arrived yet.
You are overthinking this. And missing the point.
The relevant thing is you did not get your paycheck. Full STOP.
Yes, I know you have a brain full of very complex stories that you want to spew in stream-of-consciousness fashion, as we've seen in OP and two answers so far. Stop doing that Right Now because you have no idea what actually happened. All your guesswork is unconstructive. In fact if I was your roommate and you went around town saying that about me, I'd take your next hundred paychecks because that is slander.
You don't know. It might have been destroyed in a mail-bin fire. It might have fallen out of the postal carrier's bag. It might be delayed in the mail. The roommate might be holding it for you and forgot.
Keep in mind too. Everyone you need to help you is lazy and would love to find a reason to NOT help you.
So you go to your employer (walk right into their shop!!!) and you say
I never got my last paycheck. (fullstop)
That is a serious problem and there are big consequences for cheating an employee out of their last check. They don't want to deal with a Labor Commission investigation after all. Therefore, they will want to help you.
Now consider what happens if you start running your mouth with your big fat story about your suspicions about your roommate. Nobody wants to deal with an opinionated idiot, so they'll say "sounds like you DO know where your paycheck went, so you don't need us. Best of luck dealing with him, Buh-bye."
So keep your trap shut.
Of course they will say "It was delivered to so-and-so address." And you say
I don't live there anymore.
Did you notice the part where you stop talking? Good.
They might also ask "Did you check with anyone there to see if they received it?"
I did. They assure me they did not receive it.
AND SHUT UP. Yes, I know you are exploding with the urge to tell them your whole conspiracy theory about your roommate cashing it. Shut up shut up shut up. You don't know and guesswork only hurts.
Now who knows what's going to happen? They will investigate of course, but they will want to see if you in fact did cash it. It should be apparent in their invesigation that you did not, so they should reissue it without further ado. If this check was stolen by someone, they will chargeback the account it was deposited into, which would suck for that person. This is a big bag of Not Your Problem. The system will deal with it. You don't have to be Nancy Drew here.
My employer refused to deal with me / talk to me
Then you go straight to your state's Labor Commission and file a complaint. This is pretty easy, and the employer will really, really, really hate it.
My employer worked with me, but swears I cashed the check
Then straight to the police and report check fraud. They can dig into which account it was deposited into and who owns that account, or if it was cashed at a pink neon check cashing place, the cops can force them to pull their CCTV tapes and see who stood there and cashed it. All of this is also **a big bag of Not Your Problem.
Don't Nancy Drew this stuff, becuse in the real world, people who play Veronica Mars/amiss Marple/Agatha Raisin get treated as suspects because they know too much.
At some point in the investigation, it's possible a cop will ask you "Do you have any suspicions about what happened?"
Now. Finally! Now! You get to tell your crazy story.
But the next thing he'll say is "Do you have any reason to believe this?"