Do I have to control for the different starting times (cohorts?), for example cohort1 for everyone starting in 1960-1969, because they will be more likely to have an event since they started so early?
You should certainly control for the actual starting date, but not because those individuals are more likely to have an event. The increasing probability of an event with time since the start of career is handled by the survival modeling itself.
Most important, based on my experience covering most of the time frame in question, the ability of women to have a long, successful academic career has changed dramatically over those decades. Thus you should consider an interaction between starting date and gender. Model starting date as continuous, with a flexible spline-based form, rather than breaking it up into decades. That will provide better modeling and should require fitting fewer parameters.
There also might be some changes in the structures of social networks over the years, depending on how those are defined. At least in my field, large numbers of authors per paper are much more likely in recent years than they were at the start of my career. Also, expectations for publications per year seem to have increased over calendar time, which might affect how you define "high status coauthors" or how you evaluate things like publication rates per year. Those sorts of things will require control for the actual starting date.
Am I running into some kind of bias here?
One thing that comes to mind is a significant competing risk that can lead to an end of publications: death. It's not clear how your model will take mid-career death into account. Also, you don't say how you will handle those who change gender. The case of Ben Barres illustrates both those problems. On the other hand, death doesn't immediately stop publication output; Ben Barres has been dead for over 3 years yet still has a publication that came out this year.
Another potential source of bias is differences among academic fields, and how those might have changed over time. Women have been more generally accepted into some fields than others, with a pattern that seems to have changed over the decades. Different fields have different possibilities and expectations about publication productivity, which could affect covariates you associate with individuals and who counts as a "high status coauthor" depending on the field.
If you are including time-dependent covariates in your model (which seems likely), you can also run into problems with survivorship bias. For example, my H-index probably depends as much on my having lived long enough to have nearly a 50-year publication history as it does on the quality of my publications.
These aren't easy issues to deal with. I'd suggest working with a local statistical expert from the start to make sure that the design and conduct of your study are solid.
