I think there are multiple separate events here, timed to coincide in order to minimise disruption. It's hard to be sure whether they share overlapping causes, but I'd guess that there's at least an element of that.
Yes. In public relations you never want to trickle out bad/concerning the news, you want to "rip off the band-aid and preferably follow it up with some positive/distracting news like 3.5 Pro model release or price drops.
Gemini had come under Hassabis, being co-led by Shazeer and Vinyals, who are now both gone. Gemini will now come under Koray Kavukcuoglu, reporting direct to Pichai as SVP of DeepMind.