Can OSX still sync dropbox while "asleep"? If a Mac Mini running macOS 10.11 is in sleep mode, but powered on and connected to the network, will Dropbox still sync in the background?
You might want to consult Apple's documentation about Power Nap.

Mac Minis (Late 2012 and later) are Power Nap compatible.

From what I can see, the only services which are available when a Power Nap compatible computer is in sleep mode are Apple-only:

If your Mac supports Power Nap, these activities can occur while your Mac is asleep:


Mail receives new messages.
Contacts keep up to date with changes made on other devices.
Calendar receives new invitations and calendar updates.
Reminders keep up to date with changes made on other devices.
Notes keep up to date with changes made on other devices.
Documents stored in iCloud keep up to date with changes made on other devices.
Photo Stream keeps up to date with changes made on other devices.
Find My Mac updates the location of the Mac, so you can find it while it's asleep.
VPN on demand continues working so that your corporate email updates securely. (Power Nap supports VPN connections that use a certificate to authenticate, not VPN connections that require entering a password.)
Mobile Device Management can remotely lock and wipe your Mac. 


And these activities can occur while your Mac is asleep and plugged in to an AC power outlet:


Software updates download.
Mac App Store items (including software updates), download in the background.
Time Machine performs backups.
Spotlight performs indexing.
Help Center content updates.
Wireless base stations can wake your Mac using Wake on Wireless.


Content lifted directly from the Apple Support document linked above.