It seems to me the Templar is really the only choice for healing. I'm immensely new to this game's mechanics, but will healing be focused around the weapon skills or the class skills?

For example, can I be an Earthen Knight and use a Restoration Staff to heal and support?

Yes.

Similarly, templars can invest nothing at all in their healing tree and not be healers.

Just got a whole hell of a lot more excited for this game then. I can't wait.