Points of different shapes in mapview function
I would be interesting to have a way to colour the points, but also set the shape of each point. I have a case where it would help a lot to colour the points with a palette, but to shape them differently depending on another variable. Could this be added?
Quick note: is there a way to add a ggplot graph to mapview?

There's a manual solution for your first question here. With manual I mean that you'd need to create your map with leaflet rather than mapview. HTH.
Re ggplot, this is not possible currently. Though I think it could be done. Not sure how much effort this would be and it's not high on my priority list. If it does not require much effort though, I'd be happy to provide a mapview method for ggplot objects.
I've looked into it and it is not optimized for what I want to do. Whenever I use the pch =21 and the bg in conjunction with the col, it's not plotting correctly the shapes and colours. A mapview solution would be welcome!
There's this plugin which would achive what you want I guess. I don't think there's a R wrapper for it yet, so I am tagging this for the rspatial dev day at user2020. Maybe someone wants to implement this...