So, I'm cobbling together a lot of informal understandings here, rather than a disciplined reading of Butler or other philosophers of the area. If another comrade with a more formal understanding wants to step in and correct me, I defer entirely to them. They are correct, I am trying to synthesize a bunch of stuff from a scattershot understanding:

I'm gonna short-hand some philosophical concepts here, so if they don't entirely land, I can try and follow up on them after the fact. I'm also gonna do this in bullet-point format, because I find dense paragraphs of philosophy unreadable.

• Gender is a socially constructed (which is to say: it is not intrinsic) role which is performed (which is to say: gender is a series of external behaviors and signifiers).
• This is important because it is directly opposite to sex, which is (at least for now) determined by intrinsic biological traits. What is critical to understand here is that there is nothing about gender which is inherently tied to physical sex.
• To make this theoretical model more concrete: Possessing a certain set of chromosomes and/or genitals does not, for instance, actually mean you 'can' or 'should' wear a dress. It does not (despite some claims to the contrary) make you better disposed to caretaking tasks or productive tasks. We can therefore understand gender as constructed in its Anglosphere form as the set of behaviors and signifiers traditionally assigned to individuals based on physical sex which are not in any way intrinsic to physical sex. This contains everything from expectations of appearance and modes of speech to socio-economic roles.
• One of the important proofs of the difference between gender and sex can be found in the ways in which gender varies from society to society. We can point to the existence of different concepts of a "third gender" (scare quotes because that's often a massive oversimplification) in some societies, for example, to challenge the traditional binary found in the Anglophone world.

So, now let's talk a little bit about Marxism and gender:

• First, I want to make sure that we clearly state that treating a superstructural element (and gender is mostly superstructural) as wholly dependent on and responsive to the base is vulgar Marxism. Don't take it from me, take it from Stalin (in his answer to the first question) and Engels. This is important, because one tendency of some Marxists is to treat superstructural elements as fundamentally derivative and without any motive force of their own. This is not so.
• Second, I want to emphasize that despite that fact, gender has been shaped by changes in the base that are easy to follow. Despite the way that modern conservatives emphasize the 'conservative' principle that 'women's work is in the house' or other such nonsense, the period they harken back to in order to justify this view was, in point of fact, exceptional, and their perspective is mostly bourgeois. The simple fact is that under industrializing capitalism, women and men of the working class both worked outside of the house because they could not possibly survive without doing so. We can also see this relationship in frontiers culture in the US, where the rights and roles of women were significantly expanded on the frontier long before they were expanded in the more developed east.
• Third and finally, the abolition of capitalist relations of production transforms gendered performance, but there is no reason to believe it eliminates them. This is important because there is often a call in radical circles for the abolition of gender. This seems to me, at the very least, to be a quixotic task. Rather than seeking to abolish gender, we must seek to liberate it - to free each individual from being assigned gender by society and allow them the power to choose, change, and define gender roles in a way that allows them to be more themselves, not less.