If you did a straight upgrade from Windows 7, its possible that the app manifest list hasn't been updated properly. You can try uninstalling Dropbox, repairing any broken or missing registry keys with Ccleaner, and then reinstalling Dropbox again.
I had some weird issues with Firefox, Chrome, GIMP, but these were all related to the upgrade process and some quirks with the window manager. Reinstalling them fixed my problems.
The network stack in Windows 10 is slightly different, Virtual Box may need to have an update done to fix whatever issue its having. In the meantime, create a virtual network adapter and interface with your VMs that way for now.
For anyone having issues in general with Windows 10 after the upgrade, backup all your stuff, make sure Windows is activated, then perform a clean install. If you're not up to downloading the ISO, perform a reset through Settings > Update and Security > Recovery > Reset this PC > Choose "Keep nothing" when you get the prompt.
Running the same VM on my laptop, I can switch to a bridged network. I can see all the devices on the network as well.
Could it then be narrowed down to an outdated network driver for my other machine in which I am experiencing the problem?
