For eg. is an INFJ male more cognitively feminine than an ENTJ female?
No. This involves an entire discussion on gender studies I don't want to get into. Any class on gender studies, even thirty years ago, would explain why.
I suspect gender and MBTI both play a big part in personality, but is it hard to disentangle them?
MBTI does give percentages for males and females in their official results. For example, the male INFJ population is lower than the female population, and the label "rarest" applied
only to the male population. Until recently, when ENTJ overtook INFJ as the rarest type, INTJ females were the rarest according to the official MBTI. INTJ males were more common than either INTJ females or INFJ males.
There could be a lot of reasons for this tied to gender politics, but gender politics describe a society's view of gender, not traits that occur in nature. This wouldn't be a discussion about MBTI.
The biggest flaw in MBTI is that people take the test on themselves so as long as our society views certain traits as gendered, people will choose those traits due to self-flattery. This doesn't mean, for example, that men are less empathetic or that people who believe they have empathy actually do. Societal pressures of different kinds weigh in as well. Being introverted was once considered a flaw. Now people boast about it and the newest results are showing a shift: an extroverted type is rarest now.
ENTJs are 1.8% of the population. 2.3% are men. 1.5% are women.
ENFJs are 2.2% of the population. 1.4% are men. 2.8% are women.
INFJs are 2.3% of the population. 1.4% are men. 3.1% are women.
INTJs are 2.6% of the population. 3% are men. 2.2% are women.
ENTP are 4.3% of the population. 5.1 are men. 3.6% are women.
After ENTP there is a big jump in the percent of the population for each type.
Currently, ENTJs are rarest overall, while INFJ men and ENFJ men are tied for rarest by sex. INFJ females aren't even close to "rarest".