SSDs are used in datacenters, though in my experience they haven't completely replaced spinning rust. Typically I see flash as a (large) cache layer or for more important stuff, with spinning rust for bulk storage.
The transition to various flash-based storage is essentially complete for anything that is inside the actual server (with booting the thing off SD card and that being the only storage inside being somewhat common), but for SAN systems SSDs aren't that much interesting, because the bottleneck is not in the storage media itself, but in the interconnect technology (typical SAN storage shelf can saturate it's FC ports on random IO traffic regardless of whether it is based on flash or spinning rust at 15k RPM)
Most of the storage is and for quite some time will be magnetic until SSD costs lower enough. Also am not sure but I think for long term cold storage (backups, archives) magnetic can have advantages.
Yes, for primary disks and cache tiers. Archival storage and large disks are still on HDD, usually through some SAN vendor that handles replication and performance.
They are still not cost efficient when compared to traditional hard disks, so they are usually used for intent logging and de-staging to conventional disks. The pricing is nowhere near competitive in USD/GB or in total capacity for SSD’s to replace conventional disk-based storage.