https://rive.app/ is built with Flutter. I use it a lot works very well for me. It's a good example imo of the kind of website that Flutter is a good fit for.
I assume their medium term plan is to make the new wasm backend the default rather than continuing to rely on transpiling.
I haven't seen one.
I like Flutter, but I don't think the Flutter Web design is very viable for production apps. It renders everything in canvas. Dart is already sluggish on native, and cross compiled to JS it's usually unusable.
I think Flutter will be a good choice for mobile and Desktop soon-ish, but for web ... maybe in a few years. Maybe.