No. Not exactly. Why would the dog have to be female to be considered more subservient to either a human or another dog?
Beats me, but that's a bit like asking if someone got cut off by a woman in traffic, and yelled out the window "You stupid bitch!" why male dogs are considered such better drivers than female ones
It seemed like you were trying to explore whether or not the origins of the word bitch played a role in making that insult sexist or not.
I think "bitch", by itself, gets used in sexist fashion because it's often used as a way to denigrate women who are being justifiably assertive. Going the direction of asking whether female dogs actually have worse dispositions than male dogs would seem like a fruitless exercise because it's really the use of the word that causes sexism issues, not the origin nor the way it demeans female dogs. It's a word that generates real harm because it creates challenges, especially in the workplace, for women to be able to do the same work that a man does. Like if you're a female trial attorney, you have to make special consideration for not coming across as bitchy whereas male attorneys don't really have that challenge.
"Make soandso my bitch" may have some sexist aspects to it by way of referencing a specific gender, but I don't think it's nearly as severe (I don't think it's the type of expression that changes anyone's behavior nor reinforces any gender-based power imbalances) as bitch by itself. It's an expression primarily used in competetive gaming to emphasize or illustrate a general power imbalance.