Work in government jobs, potentially sensitive in nature?
If a federal job is "sensitive in nature" then the inevitable background checks would flag anything nefarious. I'm not really sure what you're getting at here.
I'm not sure why their only living in America for one year would disqualify them from serving on a jury. If they are otherwise unqualified, they would be dismissed. And I imagine "I am currently not living in the country" would be a perfectly valid reason to be excused anyway.
The Constitution requires someone to have been a US citizen for seven years to serve in the House of Representatives and nine years to serve in the Senate, and virtually every state has residency requirements - so this wouldn't be an issue.
But if the voters are happy to elect them anyway, why should it be an issue?