"In total, 121 women and 61 men completed the experiment over the course of several months across 2013." is where I am getting the 2013 from, so if I was to say, in a study conducted in 2013, blah blah blah, very specific to the data itself, and I do an in-text citation in APA format, which year do I use?
It needs to be the year of publication in the in-text citation (same for the reference at the end of the paper). This is backed up by the Purdue OWL:
When using APA format, follow the author-date method of in-text citation. This means that the author's last name and the year of publication for the source should appear in the text, for example, (Jones, 1998), and a complete reference should appear in the reference list at the end of the paper.
If the date the survey was conducted is important then you should definitely mention it in the surrounding text like you’re doing.
Due to the fact that writing a paper then going through peer review takes time, it’s not unusual to see some delay between the date the experiment was done and the date the paper was published. Even though this does seem like a bigger gap than usual, I probably wouldn’t even question it if I was reading it.
I am not sure I am following you. If the data were collected in 2013 and published in 2017 and when the data were collected matters (e.g., prior to or following a major event that could influence the data), you would want to list both.
In total, 121 women and 61 men completed the experiment over the course of several months across 2013 ... (Smith and Jones 2017).
Adding to Laurel's previous answer: The purpose of the in-text author-year citation style is to point your readers to the full citation record of the published work in your bibliography. Consider a situation in which you cite two independent works of the same first author, e. g. Jones et al. (1989) and Jones et al. (1991), then the date is essential in associating the reference to its respective entry in your bibliography.
