If I open my mouth a little wider then usual I burp..why is this?

Dunno about the headaches but I suspect one relaxes their throat muscles and since they are connected to your chin/jaw- your mouth opens a bit wider to make air passage more manageable for your system.

yikes, how’s that for an oddly phrased sentence?

I’m not an MD, but I suspect you burp because you take in extra air. As for the headaches, may be connected to jawbone issues, like TMJ.

it’s in the genes.

Can’t resist a quip. As my literal-minded north of England husband would say, “So don’t open your mouth too wide.”