I actually agree with the general premise that nature doesn't make mistakes. I think that implies a more humanistic intent to the whole thing than is really there; nature just is. Obviously the religion element can throw a spanner into that idea but I do think if one does have those beliefs, the interpretation of God's intentions can be flexible enough to account for the realities we find down here on Earth without making you a bad Christian; but this is just my interpretation as a more or less irreligious kind of guy. If there is a divine being in charge of it all, I don't expect our feeble minds to grasp the complexities of their plans.
It's up to us to accept or deal with the consequences of messing around with ourselves and the world around us, but ultimately I think changing someones anatomy (be it through hormonal treatments or surgery) is a net good in the world on account of how much it helps the people who want it. I don't know if you know any transgender people yourself, but I can tell you from those I do know is that these treatments make them so much more comfortable in their own skins and in themselves.
Climate 'alarmists' (I think that phrases implies that people are exaggerating the threat, which I would pretty heavily disagree with) I think, for the most part, aren't saying what we are doing is 'against' nature; obviously messages get muddled but I can tell you as someone who is an actual earth scientist is that climate change is the natural consequence of pumping a shit load of carbon and other greenhouse gases into the atmosphere. Whatever happens, the world will keep turning and the big old rock we call home will probably keep being a big old rock, but the reality of it is that we are going to make the big old rock very uncomfortable not only for us but for all the other things that call it home unless we pretty drastically change what we are doing right now.