No, the US does not have laws against convicted criminals from being elected to public office, and it absolutely shouldn't have those laws. The fitness of someone to serve is ultimately determined by the public.
If you block people from being elected (or people from voting) who have committed crimes, you allow unjust laws to ferment unchallenged, and you encourage politicians to pass laws that disproportionately affect their opponents.
Yes, in some cases, that means a murderer or a rapist might be elected. But that's unlikely, I don't see the public supporting the election of a convicted murderer any time soon.
As for Manning? She did what did for the reasons we know. In my view, I'm less bothered about the notion that she violated the law by leaking secret information as I am that she did so impulsively and without care about who she handed that information over to. She's probably a good person, but her lack of care, not the fact she technically violated the law, is a greater concern here.