I have questions about these facts. Starting with the caveat that I am not challenging them as factual. What I feel is a need to know more.
Regarding prisoners--does the data show that a transgendered prisoner in a prison for females is more likely to harm (especially sexually abuse) another prisoner than others there? Does stage of transition seem to affect it? In known cases what stage of transition is required before this kind of placement?
Yes, more likely to harm. Transwomen commit sexual crimes at the same rate as any other male, regardless of how far they've transitioned.
With respect to stage of transition... hard to say. The places that have policies allowing trans-identified people to choose which ward they are placed in are also operating on a "self-identification" policy. the UK, Canada, and California all have self-id in place for prisoners, which allows male-bodied prisoners who say they identify as women to be placed in the female ward, and there is no requirement for any degree of transition.
If the policy required either full genital surgery or chemical castration, I would have significantly less objection to males being incarcerated in shared spaces with females.
Stage of transition may also affect the question of sports--as a potential future discussion, I wonder if something related to hormone levels may ultimately need to come into play, but competitive sports is definitely one of the stickier issues and I don't know that I have the answers, so I have no directed questions either.
It varies. Some sports don't have as large a difference by sex - like dressage or other competitions of horsemanship or animal training. Some sports can be mitigated to a degree by hormone therapy - removing testosterone from the system can reduce many advantages that depend on lung capacity and stamina. But even after full transition, some differences remain. Males are larger, have denser bones, larger muscles, different hips (which affects running sports) and a host of other things. So for some sports, even after several years on hormone therapy, advantages still remain for males.
There is an argument to be made for people who take cross-sex hormones before the onset of puberty, because they will never develop the bone density and most other characteristics that affect athleticism. But that's another whole topic of debate, given that puberty starts around 12 for most people. There are concerns around consent and safeguarding when it comes to hormonally transitioning minors. That's outside the scope of this discussion.
I think for the most part, many people (including female people) are willing to accept transwomen who have undergone hormone therapy to reduce their testosterone levels to the normal female range. For professional sports, that's reasonable. For middle school, high school, and collegiate sports, that's a more complicated matter.
Regarding rape shelters--can you tell me more about this? I do find it alarming and I would like to read more.
https://www.womenarehuman.com/woman-in-homeless-shelter-threatened-by-male-transgender-resident/
https://www.womenarehuman.com/male-transgender-boasts-of-harassing-women-in-crisis-shelter/
https://www.womenarehuman.com/male-trans-pedophilia-rape-history-sexually-attacks-women-at-homeless-shelter-christopher-hambrook/
https://www.womenarehuman.com/women-in-shelter-forced-to-shower-with-man-who-identifies-as-transgender/
There's also a host of rape and domestic violence shelters in the UK and Canada that have lost funding because they are designated as "female only", and refuse to grant access to male-bodied people.