Why does my Droplet show almost 100% disk usage even after attaching a new volume?

Adding a volume to your Droplet doesn’t increase its main disk. Instead, it adds on a second disk as a second partition. To expand the Droplet’s main disk, you need to resize the Droplet to the next plan.

If you need additional storage, you can either add a new volume to the Droplet and move some files to that volume, or configure all the logs in /var/log/ to save directly to the new volume.

No; however, you can add another layer onto your Droplet with an attached Volume.
Snapshots of Droplets are a best estimate based on the disk usage. Snapshots of volumes operate at the block storage level, so the snapshot size may not match what the filesystem reports.
You cannot currently download DigitalOcean backups or snapshots, but you can use third-party tools to save your data locally.