You don't need even to download the cover (at least with Firefox). I often use 'Copy image' from right-click menu and then paste it on the old cover in Edit Metadata window. Change is immediate (if you don't click 'cancel
).
To see the change in library view it is sufficient to click on some other book and back, no need for restart.
This takes care of Calibre cover.
I have a feeling that if you convert and add first page, then sometimes you can end up with two covers. Just something to check.