Oda isn't always great when it comes to designing female characters so the evidence with those girls is just bad evidence, and also Doflamingo's jolly roger isn't upside down and his jolly roger is crossed out. In fact you could also say that jolly roger looks similar to the Heart Pirates jolly roger, and Law does know his medical science.
So in truth it was Law that secretly poisoned Whitebeard not Doffy, and now Law is targeting Kaido.
Why else do you think he's called the Surgeon of Death.
Btw as a Doffy fan even I think that whole theory of Doffy poisoning Whitebeard is a bit of stretch, I won't deny it being interesting though and it's still a bit more better than some other horrible theories out there.