Yes, I really do want you to explain what you think women having careers has to do with socialism.
I'm kind of at a loss when it comes to answering this because it seems so obvious to me. I'll do what I can...
Apologies if some of this seems condescending but since I don't know what is common knowledge in your area, I'll state things most people in many countries are aware of.
It's a big topic and I'm not willing to write down some kind of essay.
I'll be simplistic but hopefully not too simplistic. Things will be omitted. I'll be wrong and misleading about what I consider to be details for the sake of brevity but hopefully the general drift will not be too far from the mark.
I think distinguishing between these three broad categories of relevant socialist action will be useful to organize the matter:
a) social war against the old order
b) by-products of economic policies
c) accomodating motherhood
Category a) was more important historically whlie c) is more important now.
a) Socialism being a direct attack on the power of property and property being overwhelmingly patriarchical at the time, socialism was orginally basically an attack on patriarchy.
Socialists were ideologically committed to the abolition of patriarchal institutions including marriage and the nuclear family in the case of the more radical factions. This was partly due to abstract considerations such as likening women who did not control their household's property to proletarians and husbands to capitalist exploiters. But it was underpinned by pragmatic considerations as well.
The core male constituency of socialists were urban workers who in contrast with people who worked a plot of land or could afford a bourgeois lifestyle did not have the opportunity to control much of their wives' lives and also badly needed income from their wives' independent employment. A strongly patriarchal lifestyle made little sense to them and restrictions on a woman's ability to be employed were against their interests.
Women employed in mid-sized to large businesses were also a constituency of socialists while servants and women whose life was controlled by their husbands such as those working on a family farm were a consitutuency of theocrats ideologically and pragmatically committed to patriarchy, especially in Roman Catholic areas. Socialists were virtually at war with conservative theocrats and therefore stood to benefit from women having lives outside of homes and churches.
For similar reasons, socialists also stood to benefit from female literacy and secular education.
For these reasons and more, a tradition was established in the socialist movement to support policies which would directly and indirectly help women have careers and more generally participate actively in society outside of the home. This tradition did not entierly die when much of what I described above became history.
b) All women have to face what remains of patriarchal culture in the labor market. Women who choose to have children have to face an additional problem which is becoming relatively more burdensome, especially if they choose to do so at an inconvenient point in their career. Assuming this as a given and assuming away regulatory reforms and whatnot, the actual impact of these issues on gender inequality and women's career is going to depend on economic vagaries and in particular on the unemployment rate.
The easier it is to land a new job, the easier it is for women who are trying to balance the demands of their family life and their employment to bargain for convenient working hours and the like or to temporarily quit the labor market in order to take care of young children.
Among other economic policies affecting women, socialists strive to eliminate unemployment, to compensate adequately those who are unemployed (even deliberately) and more generally to strengthen the bargaining position of workers. Fully socialist regimes normally have little unemployment because the owning class (if it even exists) has no say in the matter and there's usually no shortage of work to be done with minimal inputs other than time and effort on the part of the workers.
As a result even outside of any policies specifically designed to help women get good paying jobs, socialist economic policies happen to disproportionately benefit women and especially mothers. It if wasn't for patriarchal culture, that effect would be considerably less important but that's how things stand.
c) Because women will continue to have children for the forseeable future and because economic rationality will cause mothers to face discrimination in the workplace outside of any patriarchal prejudice or harmful intent, this issue is becoming more important as gender inequality wanes.
In part due to the history mentionned above, countries in which socialism is strong or has been strong at any point since the last reactionary upheaval tend to give more effective privileges to mothers, sometimes dramatically so.
In addition, these countries tend to have cheaper and better child care and education with hours designed to make it practical for parents to hold jobs, in part due to socialization of the sector. This naturally helps parents and especially single mothers.
Outside of a patriarchal cultural context, fathers also exercise some of the aforementioned privileges so it's worth putting them into two categories: those regarding pregnancy and breast-feeding on the one hand and those regarding parental paid or unpaid leave which can be exercised by both parents. But even if such a priviledge is exercised by the father, mothers benefit indirectly (and vice-versa in principle, except that any remnant of patriarchal culture puts more burden on mothers who therefore benefit more from any relief on average).
These priviledges are also noteworthy now because they're cheap compared to their impact on people's lives, because they're particularly useful in a poor economic environment and because many countries do not have a socialist government anymore but have a socialist legacy which is now defended by moderate feminists as well.
Hopefully that answers your question.
If the above is satisfactory, I may try to give satisfactory answers on to the other topics you raised which aren't straightforward either. Short answers in the meantime:
-I'm not sure what you mean exactly by "opportunity" and "limits" but the actual outcome of German reunification was broadly as I stated (and was easily predictable)... but probably not for the reasons you might imagine considering the words you used. Also the #1 political job has been held for years by a woman from East Germany (childless, yes but still remarkable) which goes to show how weak the relationship is between average outcome and representation at high levels (same deal with Obama I guess).
-Yes, the GOP is extremely bizarre on virtually all issues.
-No, women are able to make reasonable and informed decisions about vasectomies and whatnot. These matters are trivial compared to female reproductive issues anyway.