"Gender critical" is a dogwhistle for anti-trans. Dawkins falls squarely into that camp, of questioning everything that trans people, and experts on the subject have to say. The problem with being anti-trans--or, one of the problems--is that by its nature it assumes that there's some kind of sex role in society. E.g., women are A, B, C, and men are X, Y, Z, and you can't move between them. That kind of gender-essentialism is fundamentally socially regressive. Dawkins is also quite significantly culturally Christian, and Islamophobic, e.g., he is entirely critical of Islam both as a religion and as what he perceives to be a culture, but doesn't direct the same types of criticism towards the Anglican church that he grew up surrounded by.

how you identify/feel is a different matter in many people’s eyes.

The way I see it, it's just not my business. The only time that someone else's gender identity is going to matter to me is if I'm potentially interested in dating them. They're not harming people, they're not 'taking' anything away from cis-women (or cis-men, for that matter), so why should it be my business what they feel they need to do with their body in order to feel comfortable in their own skin? I came across this a few weeks back, and it really drove that home to me.

But does that alone make him Right Wing now?

It's a spectrum, like all things. You can be pro-science, but also still have a very socially conservative view on the 'right' place for people in society, or still maintain false beliefs about the 'rightness' of capitalism, imperialism, and so on. And scientists are still human, and prone to the same cognitive biases as everyone else.

Being right wing doesn't necessarily mean being religious. Being left wing doesn't necessarily mean being atheist. Yes, that's more often true than not, but I think part of that is that the right in general uses an appeal to tradition--which includes religious practices--as part of their package. But, on the other hand, Jesus, as depicted in the 4 gospels and the early Christian church, would have been very comfortable to socialists, as would the teachings on tolerance.

In re: podcasts, the only explicitly atheist one that I still listen to is The Friendly Atheist. I find Jess to be annoyingly hyperbolic most of the time, and she frequently makes wildly overbroad statements, but Hemant Mehta is pretty measured overall. Mostly I listen to politics (FiveThirtyEight), 2A podcasts (A Better Way 2A, the irregular Tiger Bloc Podcast, Guns Guide To Liberals, Practical Shooting After Dark), a couple of ex-Mormon podcasts (Mormon Stories is great if you love really, really long form interviews), You Are Not So Smart, sometimes BtB, Cool People Who Did Cool Things, a couple others.