We accept a U.S. Department of Defense Form DD-214 that indicates an honorable discharge.  If you don't have one I can tell you what other documents we accept as proof.
will I have to keep proving my service everytime I have to do business with the DMV?
After you receive the veteran designation, it will automatically be on all of your future DMV documents when you replace or renew.
what is the refund fee we get that I heard about?
It might be the one about applying to have the Veteran Indicator added to your license, permit or non-driver ID and if you were charged a document fee, you are entitled to get a refund for the fee
Do all these new IDs and licenses have rules on how they can be used when crossing the border into Canada?
Yes, you can learn more about Enhanced , REAL ID and Standard documents on our site. I can link you to our page so you learn more about U.S. Border Crossing from Canada, Mexico and some Caribbean countries
can I do my new ID online if I want both the new real ID and the word VETERAN printed on it?
Do you have a Standard license and want to change to a REAL ID or Enhanced ID  and also add your veteran status?
yes