You will need to purchase the Full version on Steam to gain access to same content.

If you purchase the Starter edition, you will only get access to Heavensward while playing on PC.

One thing to note on PC is that, Steam and Windows PC (SE Store) versions of the game are not compatible, that means you cannot add a Steam account to an SE account that already has the Windows PC version registered.

Yes, you will need to buy Dawntrail on both platforms, if you buy it on PS5 you can only access Dawntrail areas* on PS5, you can still access the jobs on another platform without Dawntrail.
You will need to logout in an area that the other platform can access too, otherwise it wont let you login since you'd not have access because you didn't say, buy it on Steam.