Yes (200k–300.000), that’s why it says pre-humans…we didn’t arise out of nowhere, it was a continuous evolution and it seems like if those had died out we wouldn’t be here. (However, that’s not settled, there are substantial reasonable doubts over these results as hinted at with “While alternative explanations are possible” and elaborated in the other comments here.)

Good question, it wasn’t a warming and even if it was, I don’t think it can easily be translated to today’s climate change. They refer to the Early-to-Middle Pleistocene Transition (not much info at that page though). If it’s linked, that doesn’t mean it caused it – I think people in that regard far too often think of (especially singular) causes instead of contributors within a complex interconnected set of causal factors. Maybe you’re interested in this non-included paper from the same month which projects an upcoming large sudden population decline – it’s just not substantiated and one can’t just compare modern humans with other animal populations.