perhaps not on mobile chrome (it's been a while since I've tried) but certainly on mobile firefox and mobile samsung (which is chrome rebadged).

I'd not say - go and build this as an extension on mobile; but if you have the code built to run as an extension, it should be fairly quick to repackage it as a standalone mobile app, i.e. get a dedicated instance of the browser that runs your app).

Another successful example of this model is LastPass - they deliver the functionality via browser extensions and mobile apps, but they handle all the payment and marketing and CRM via their own product site.