Not all. The mayor of Dallas does not represent a party. Which made it curious when the current mayor switched from a Democrat to a Republican.

The majority of large cities have "non-partisan" mayoral elections in the US. But most politicians are still going to be members of parties. About 2/3 of the 100 largest cities have mayors affiliated with the Democratic party, since support for Democrats is much stronger in cities than in rural areas. Some states, such as Florida and Texas, have large cities with Republican mayors, but northern cities usually elect mayors who are Democrats. In partisan mayoral elections, the parties hold primaries and voters select the candidate they want their party to endorse. In cities with large Democratic majorities that also hold partisan elections, the primary is functionally more like a general election. So basically different cities function differently, but mayoral candidates are still going to have ideologies that align them with parties, and those ideologies usually have local, not just national, impact. And informed voters will still want to know those ideologies.

No but kind of yes, there are partisan elections and non partisan elections. The Hatch Act (which applies to every federal employee including a letter carrier) states they cannot be involved in any partisan elections. Which currently means almost every election now.

Non partisan elections used to be much more common in the past, but the major parties over the years (mostly Republicans) pushed it to where there are very few non partisan elections. Even school committee seat elections in my town are partisan, and because of that my local letter carrier couldn't participate even with their massive involvement in the community.

Yes, effectively every office from Sheriff to even judges. If there is an election the candidates list their party affiliation. Sometimes resulting in funny stuff like this: https://apnews.com/article/e1580367018108b09755dfb994395c0a