The short answer is, yes, it will do exactly what you describe. You can even map different folders to drive letters (we have one folder that is called Photo Gym, mapped for drive letter G, another named PG clients mapped to drive letter W).
Once you have connected your Synology to your router you can go to “findmysynology” in your web browser and it will usually find it on your network and tell you the IP address. Use that IP address in File Explorer to see the folders installed by default. Then you just create folders, navigate about, map drives as you would with a drive directly connected to your computer.
You should also check out the Cloud Sync app. It will let you sync whatever is in your Dropbox account to your Synology.
Lemme know if you have other questions!