Exactly. Mihawk extreme diff.
Yes it does. Fujitora uses gravity to enhance his swordplay. By this logic, a swordsman who uses Haki isn't a swordsman, they're a Haki user who uses the sword.
Kaku was a martial artist and had a DF, and was still a swordsman.
Doflamingo never got scared of Kaido. He was pissed off at Law because Law was disrespecting him, the thing he hated the most. Doflamingo has a very severe superiority complex, which is why he became enraged at Law for attempting to blackmail him. It had nothing to do with fear.
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Last I checked, fear isn't represented by turning red, snapping and seething.