No. I never use the YouTube app on my iphone. I always play a video by doing:
Chrome app -> youtube.com (no prompt to install/switch to YouTube app, simply redirects me to m.youtube.com) -> search video -> play (which plays out of chrome and does not switch). The whole process is exactly how I'd hope/expect it to be.
I've exclusively used Youtube through the chrome app instead of the Youtube app because a few years ago I found that the search results were significantly worse/more heavily filtered through the Youtube app for some reason and never used it again.
Yes. If you have the app installed infuriatingly persistent, less so if you dont. Not having the app installed means you have to deal with the terrible youtube web player (which doesn't even use native fullscreen!).