How to Reduce SSD Wearout in Proxmox?

[Impact]

How do I check if Proxmox can even see the HDD?

node > Disks or lsblk -o+FSTYPE,LABEL,MODEL.

How do I format it (and which filesystem should I use for a single HDD like this)?

node > Disks > .... What to use depends a bit on the use case and model number of the disk. Please share yours.
I like ZFS but it tends to perform badly with VMs on HDDs. without disabling sync or having a separate SLOG. CTs are fine.
LVM-Thin would by my secondary choice but you can't directly store files there like you can with ZFS. I don't recommend Directory for storing guests.

How do I add it as a storage target in Proxmox so I can actually assign VMs or backups to it?

Above way does that automatically if selected. You should edit the Thin provision option for ZFS though. You can create/edit via Datacenter > Storage.

Any mount options I should use to keep things running smoothly?

This is too vague and depends a lot on storage, guest type, etc. Something I recommend most often is discard. Not as mount option though.
I'd suggest relatime instead of noatime which, AFAIK, is also the default for ext4 nowadays.