This depends on the state. Being convicted of a felony in Florida bars you from running for public office or voting in the state (until granted clemency). A conviction in Georgia would similarly bar Trump from voting in Florida, but not one in New York (which does not disenfranchise felons).

But yes, it doesn't affect his ability to run for federal office, even if it would make it hard for him to campaign from jail.

The vast majority of american democracy relies on "norms" and "the public will do the right thing" and "surely they will only elect people who deserve it"

For the vast majority of elected offices, one of the most important qualifications is "registered to vote in the district you are going to represent". Most offices have some age requirement (which might be as simple as "old enough to register to vote") and extremely few had some occupational requirement (like District Attorney has a law degree/license requirement, but Sheriff has no such requirement). When I ran for office in Colorado, failing to be registered to vote in the district you represent was the fastest and most automatic way of getting thrown out of office: no trial, no impeachment, just out you go.

The requirements for President are listed in the Constitution. Getting on the ballot in each state is harder as each state has different requirements.

You know, we also have this thing in the Western world called "the rule of law". It's this really weird idea that when someone breaks the law, they should be punished for it, no matter how powerful or well-liked they are.