You don't need to register your drone with DJI, you only need to register with the government (in the US).
Your warranty goes into effect when you activate the drone.
Yeah, SF is very funny about their personal article policies. I filed a claim for a stolen Manfrotto tripod with a replacement value of about $350. After the claim was settled, I got a call from my agent who, in a roundabout way, said: Don't file...
Yes, you're right if you're only insuring one or two items under the personal articles policy, they are inexpensive. What I find annoying with my carrier is that I have multiple policies with them. You'd think they wouldn't get upset over a $350...