I believe in reincarnation because I have always remembered. So what I have to say is based on a very deeply held belief however I am fine with people having different views or simply not believing me - I'm not fragile
When I was a young child (6 or 7) my father tried to 'teach me' the current biblican Christian version of Heaven/Hell and Adam and Eve. I found it utterly ridiculous and that's when I realised that not everyone remembered and I should keep my mouth shut. I am not a Christian nor do I belong to any other religious group, however there is plenty of evidence that reincarnation was part of Christian theology before those in power decided to cut up the bible to fit their political ends.
Also, 'past-lives' isn't entirely accurate since past, present and future don't actually exist on a soul level, it's just the material world that conforms to linear time. But that's getting a bit esoteric.
I went through a period in my early 20's when lots of 'information' about my previous lives came to me, but I then realised that remembering isn't as important as it may seem. Following your inner truth and having faith in that Truth
or the Universe or God or whatever word you choose to use, is what is really important. Also, with knowledge comes responsibility, and believe me, you may not want to remember some of the shit you've done!
In terms of gender, yes I believe that you can be attached to the gender you were in your previous life and therefore feel you were born in 'the wrong body' if your sex/gender don't 'match'. However, that in itself is a learning experience. We all have free will and can live how we choose. Karma is not 'punishment', I think this idea is from the western view of an angry god, Karma is just 'what comes around goes around'. Also, we have forever to work things out (since there's no 'time' anyway) so there's no need to stress.
Also, Garconne, I have also recently read books about children remembering past lives and find it fascinating. When I was 20 my housemate had a young child and I remember once asking her if she could remember where she was before she went in her mummy's tummy. She did! She told me how she died in her last life. It was very matter of fact and she had that 'calm wise' look about her as she was telling me. I didn't take it any further since she wasn't my child.