Call up Home Depot, other places that sell water heater or plumbers. Some places, like Home Depot, will give you a quote on-line.

The atmospheric vent (flue) is just 4" sheetmetal duct. As long as it is performing, it looks just as old and dirty as the walls and top of your green expansion tank. That doesn't affect performance. If any piece is rusty, you can replace them with the same.

The new water heater should come with a new draft hood (collar on the top that sits on "legs") and that can just be inserted into the existing elbow.

This is based on the assumption that you have purchased the same height water heater.

Switch to tankless, since it's gas. You'll save money after about 2 years.

The atmospheric vent, as @3onthetree mentioned, is a simple 4" metal vent.

They do "rot" however and need to be replaced at times. You will generally see this on the bottom of a horizontal run.

Take a pick or a small Phillips screwdriver and see if you can poke a hole anywhere you see corrosion, again particularly along the bottom of that horizontal run.

The reason it happens is because the flue gasses contain a lot of water vapor that can condense on the metal flue. That condensate is acidic, and will eventually eat through the sheet metal. It's very common on 80% furnaces, but also on atmospheric water heaters.