I know gold, stash and a few other things are shared among characters.
Is this going to still be the case with seasons? Will this stuff be shared between eternal and seasonal characters?
Or maybe some, but not all, shared? Like gold shared but not stash?
Probably and hope not.
I have never seen a season where you could keep your ressources and gould you collected pre-season in the seasons too.
No, they will be shared between seasonal characters, but not between seasonal and eternal
It is not shared but at end of season you get it all on external
So after 4-5 seasons our eternal characters will be very rich. 
It doesnt in any seasonal game i know, so 99% sure it wont
Well yes.
I have billions laying around in my D3 stash after so many seasons.
You will be just as rich if you were to stick to playing eternal realm characters during the duration of seasons… its just an illusion because you sending leftover gold from the season. But in general same rate besides maybe little bit faster with the smouldering ash buff.
They want us to all start fresh. So we all got to be poorzies.
You will be more rich if you play eternal because you don’t have to keep rebuying gear from the discord auction house since that is the only way to reliably find good gear
It wasn’t in D3 so I would suspect the answer is no