calibre's ebook-viewer does not support highlights or notes.
It does support bookmarks. But ATM there is no sync capability.
Note: The Content Server is currently undergoing a major rewrite, during the course of which it is expected that an in-browser viewer will be added.
If I had to guesstimate, it will be completed "sometime in the next couple of months".
At that time, Kovid intends on adding notes and highlights capability as well, and the ebook-viewer will be much more feature-complete.
Once that is released, I do expect someone will write a plugin to sync that with the Kobo.