Is it possible to create a new note soley by logging into iCloud.com? Also, though I can see my existing notes synced up there, I cannot edit them. Is this the default behaviour or a bug?
icloud.com does not have a notes feature at this time. It is rumored that around the time that iOS 6 is released, icloud.com will be updated to include notes and reminders. Previews of these features might be available to you already, if you are an Apple registered developer and have updated your iOS device to an iOS 6 beta, by visiting beta.icloud.com
It's worth noting that you're required to have an iCloud email address for the notes syncing. If your notes are set to sync with another address (such as Gmail), the web app can't access them. I believe this is because the Notes syncing still uses IMAP behind the scenes (each note is just a mail message in an IMAP folder).