The Old English word for pebble is papol- from papolstan. In Doric, psephos is psaphos. Ok, they are probably not cognate, but I haven’t found anything to say otherwise… I can understand that Latin ‘habere’ is not cognate with the English ‘have’ because it’s well documented, but this is frustrating because of the scant information…
I think your instincts are fundamentally right.
As for habere vs. have — they are eventually[/u] cognate, if you go far back enough into proto-Italic and proto-Germanic — it’s really not a coincidence. The same is probably true of pebble.
P.S. Love the Optimus Prime quote!