What is the government?

Small communities, mostly physically isolated from each other, but informationally connected and centrally administered. Basically meritocratic in structure - pass enough of the tests and you can work for the gubmint.

How does it decide and enforce allowed research, and allowed self-modification?

Virtually all sophisticated equipment is communally owned and equipped with government-designed protocols. Key goes to the library for eir computer time because ey doesn't have anything more sophisticated than a toaster in eir house. This severely limits how much someone could autonomously self-modify, especially when the information about how to try it is also severely limited. The inconveniences are somewhat trivial, but you know what they say about trivial inconveniences. If someone got far enough to be breaking rules regularly, they'd make people uncomfortable and be asked to leave.

How does sex-choosing work?

One passes some tests, which most people manage between the ages of thirteen and sixteen, and then goes to the doctor and gets some hormones and some surgical intervention to be male or female (or some brand of "both", and some people go on as "neither" indefinitely, but those are rarer).

What is the society like?

Too broad for me to answer - can you be more specific?

Is Key forced at a certain age to be in some regime, like our schools? If not, are there any limits on what Key or her parents do with her life?

Education is usually some combination of self-directed and parent-encouraged. Key's particularly autonomous and eir mother doesn't intervene much. If Key did not want to learn anything, eir mother could try to make em, but the government would not help. If Key's mother did not want em to learn anything and Key did, it would be unlawful for her to try to stop em. There are limits in the sense that Key may not grow up to be a serial killer, but assuming all the necessary tests get passed, ey can do anything legal ey wants.

Thank you for the questions - it's very useful to know what questions people have left after I present a setting! My natural inclination is massive data-dump. This is an experiment in leaving more unsaid, and I appreciate your input on what should have been dolloped back in.