Could shutting down the government somehow be an election stealing tactic by Republicons? How would that impact the government's ability to ensure fair elections.
The elections are run by the states, not the federal government. Congress keeps working during a shutdown. I can't see how it would hinder the election.
It would have unpredictable effects on the election though, so I can't imagine they really want to risk it.
On January 6, yes, but they don't stop working. The shutdown only affects the rank and file government employees that do things like run the government agencies, air traffic control*, Smithsonians, etc.
*Air traffic controllers would be considered an essential function, so they would actually get the joy of continuing to work without pay, until a budget is passed, then they would get back pay. (There's also a possibility of some agencies having money, for various reasons, to continue to pay essential employees.)