A few generalities:
Possibly the best way to split the difference on a single social image to serve multiple channels is to go 1:1. Twitter technically also works well with 16:9 but FB does not; so in general it's a good idea to stick with 1080x1080.
This same image size should work for your Favicon, though I would stress that given how small favicons show up, please keep details to a minimum and contrast high, otherwise people won't be able to understand it. I would keep that very top-level logo stuff in terms of design. Like, not more complex or muddied than what I have in my avatar here!
You can see a wide cover image here, at the top of the page. Beyond 21:9 I don't recall the exact dimensions, but you should be able to grab that one and examine yourself.
I don't know on logo, sorry. Hope this helps though!