Unfortunately, there is very little research on most of your questions.
Some of it isn't being done because it would never be funded in the current day as even asking these questions is considered bigoted. Which is weird, considering similar studies have been done on homosexuality.
Some research on things like the rate of regret of transition, detransition, or desistance, have been actively suppressed by transgender rights activists.
"- the percentage of transgender people who were born as men and now identify as women by comparison with those who were born as women and now identify as men."
It used to be approximately 5-10 times more common in the male -> female direction.
Now among young adult in the UK at least, it is about 3 times as common in the female -> male direction.
"- the difference between having gender dysphoria and being transgender (and how much of an overlap there is)"
There is no agreed upon consistent definition of transgender so this question would be answered differently by everyone.
Some transgender people say that gender dysphoria is, essentially, a diagnostic criterion. If you experience it, you are trans, full stop.
Others (the more radical activist wing) put it in terms of Gender identity theory which basically means you are whatever you "feel like" even if you are physically your biological sex have no medical intervention and have no issues with your body, if you "feel like you are actually female" then you are transgender.
Many gender critical people and some therapists would say that dysphoria around one's sexed body in fact can occur without an individual being trans. For example, if someone has experienced severe abuse they might disassociate from their body and wish it weren't the sex it is. Or, they might hate their body for reasons of internalized misogyny. Or a man might come to hate his adult male body if he associates it with a male relative who was an abuser. etc.
The dysphoric feelings of people in these situations, are unlikely to be helped by medical transition. Yet it is now considered "conversion therapy" to help people work through such issues if it results in them becoming more comfortable with their natal sex.
"- the percentage of children and young people who have temporary gender dysphoria i.e. they have a period of time identifying as a gender that is not their birth gender but then revert to their birth gender as adults"
There is no data on this among recent cohort of trans children. So it's hard to say. Previous data giving the approximate 80% figure specifically for boys identifying as girls is a bit dodgy in my opinion because it did seem to include some gay kids whose parents were hoping to cure them. Also the earlier studies did encourage parents to force their kids to gender conform (e.g. "wear boys' clothes" etc) as a mode of treatment which isn't appropriate.
In any case a large chunk do desist. No one seems to be able to tell who will and who will not.
"Then, specifically, on transgender women in sports, what are the guidelines:
- do you have to have had gender realignment surgery in order to compete as a woman (if you were born a man) or vice versa?
- do you have to have lived as a woman (if you were born a man) or vice versa for a specific period of time in order to compete in that gender group? And if so, what is the period of time? "
No surgery is needed but for adults in international competition, males must have their testosterone reduced below levels set by the agencies (either via surgery or hormone treatments).
Many youth sports agencies are now allowing people to compete cross-sex with absolutely no intervention of any kind. The argument is because it would be unethical to require youth to damage their bodies in order to compete with their gender identity group. Of course they could simply compete with other members of their biological sex however they identify.
"- If someone has transitioned to being a woman with surgery, hormone therapy, lived as a woman for years etc how big, really, is the competitive advantage (bearing in mind the fact that athletes born as women come in all shapes and sizes and with bodies that are more or less suited to athletic activity etc)?"
From personal experience, I think it would still be very large. My husband was transgender and on hormone therapy / T blocking drugs for over two years and was still MUCH stronger than me by the end of it.
I suspect that the average trans woman (male at birth) will certainly be stronger/faster/etc than the average female person, though probably less than the average male person.
However, among elite athlete there could be a larger advantage for transgender women (males). This is because most of these men before they transitioned actually competed with men so in order to have gotten to that position, must already be much, MUCH stronger/faster than even elite females. Thus when they transition although they will get slightly weaker, they will still be much, MUCH stronger than the elite female athletes.
However no one has looked at a large enough population of transgender women to determine this. I certainly hope some physiologists do this research because it would be very helpful to the IOC and other sporting groups.