Uninstalling the app

If I uninstall the app from my Shopify store, does it remove all the code that was added to my theme/website?

Auto Tags doesn't make any code modifications in the store in the first place so when you uninstall the app, no code is left behind.

If I uninstall the app, will it remove the tags applied by it already?

No, we don't remove any tags if the app is uninstalled. The only time we remove tags is when a customer explicitly configures a workflow to remove them.