0240.2 Assessment
Research Question (Clear, unambiguous) "Does female political engagement promote legalization of same-sex marriage?"  This sounds reasonable on the surface, but becomes less clear when we look closely.  Political engagement is very ill-defined.  Does this refer to the proportion of women who vote (compared to men) or the proportion of elected officials who are female or something else?  Since legalization of same-sex marriage is largely only a recent issue but women's political participation goes back much further, the question needs more specification.
Relevant Literature There is a fair size literature on gender differences in support for political issues that is relevant.  There is also a (smaller) literature on the impact of women in legislatures.  These are probably more important that the narrower literature on differential support for gay marriage by gender.
Causal Interpretations (clear, all parts defined, mechanisms, controls, plausible) The general question of how and why same-sex marriage went from anathema to legal support in a short time is difficult.  Teasing out a possible role of women's political participation may be hard.  To do so, you will have to develop a more complex and nuanced set of causal possibilities than shown here.
Data (variables, sample, comparison) The apparent plan, from the data listed, is to merge data into a set coded by country and year with the proportion of female legislators and the legal status of gay marriage.  Presumably this would be assessed with some kind of event history analysis or the like with the intention of testing if countries with more elected women accept gay marriage earlier.  This seems a reasonable approach at this level of analysis.  It is not clear if this level of analysis is ideal, but that is a different question.
Research Value Omitted in project outline, but the value seems evident (unless someone else has already done something like this).
Overall It is not clear to me that the main question is theoretically likely or empirically answerable (given the historical realities).  However, the question is a plausible one and the research design seems a workable way to seek an answer.