You can create a contact and specify email, but the only option for updating a contact seems to be name. Is it possible to update a contact's email?
This would be preferable to making the user delete the contact and then add it again it with a new email, in cases where they mistype the email (or if the address changes, I suppose).
Nah, there's no way to change the email address. You can programmatically delete the contact rather than making the user delete them, I think this is what the website did.
Just a note, as I mentioned on this other thread (Add contact to pushbullet with the api): the official apps use the (not yet documented) /v2/chats objects instead of contacts