Fullscreen will hide the window decorations, but that won’t solve the use case of “behaving like a desktop application”. I use PWAs for websites that are applications (Outlook, Teams, Spotify etc). I want these windows to be dedicated to those applications and nothing else. They should appear in my window list on alt+tab, not be able to navigate away to something else etc.
Yes, if you want it to. There is a pwa button that appears on pwa supported sites that lets you toggle between app window mode and normal browse tab mode.