First of all: Welcome to the Storj Family
I think the Synology Nas handles that for you, just verify it with a reboot if the Storage is on the same mountpoint as before (/volume1/ and the other disk on USB).
Yes, as long as docker gets also started automatically as a Service (which i assume on Synology) all containers will also be started. Thats because of this Flag (which is a RestartPolicy, see Start containers automatically | Docker Docs): --restart unless-stopped
To be 100% sure test it with a reboot of your NAS
As long as you configure it as in the Documentation yes, i never had any issues with it. But to be sure i would recommend to verify it on a monthly basis.
Also i would recommend an external Monitoring tool (like uptimerobot), so you get alerted in case something goes wrong. Also you might want to have a fancy Dashboard with Historical Graphs, check this Guide out: