I've observed the effects on both Firefox and Chrome and it's like day and night.
On chrome you can even test it using Devtools > Application> backward forward cache
Or Vimium users could RTFM.
Plus in the case of your parent, their website implements vim keybindings, so basically all of them are going to conflict with this extension, and it's going to be pretty useless if they detect that situation and react by changing all of them to something non-standard.
No, not really. Not unless it specifies certain `web_accessible_resources` which you try to load - but that's a bit of a hack, needs to be done on an extension-by-extension basis, and won't even work for all extensions.
The initial way we checked for it was looking for document.body.matches('*[_vimium-has-onclick-listener]') but apparently that stopped working in 2018. We then used a variation of this code: https://github.com/hackape/Detect-Vimium/blob/master/detect-... which seems to work still.
Vimium disables the browser's backward-forward cache (by listening to the "unload" event), causing navigation to be much slower.
There is also no way for a website owner to warn a Vimium user of conflicting keybindings. I run a website which implements Vim keybindings and several of my users complain that it doesn't work. I have to remind them to disable Vimium. This issue has been open on their GitHub for more than 4 years with no end in sight.