The most common issue I experience with Genymotion is inability to start a virtual device. At least now I believe I've figured out how to fix this behaviour.
It seems that Genymotion waits for a short time for the device to get an IP address from the VirtualBox DHCP server that serves the Host-Only network adapter. To work around this issue, the virtual device can be started from VirtualBox directly. Then, once the IP is assigned, and this is visible on the screen, the device can be started in Genymotion. It will simply continue the already-running session and will display the graphical screen in the Genymotion window while the console output will remain in the VirtualBox screen. One could also run the VM headless, in which case there would be no VirtualBox screen open, I guess. That way it would be harder to figure out when the IP address has been assigned.
Another issue that may cause this behaviour seems to have to do with the configuration of the DHCP. The main reason here is the discrepancy between the DHCP address range and the virtual network adapter IP address.
Sometimes there are multiple adapters in the Host-Only section of the VirtualBox network configuration. Make sure there is only one. Remove the others.
Then, synchronize the IP range of the DHCP server to match the given IP address of the host-only adapter.
Example: if the host-only adapter has the IP settings 192.168.248.2/255.255.255.0, the DHCP server can be configured as such:
Server Address: 192.168.248.100
Server Mask: 255.255.255.0
Lower Address Bound: 192.168.248.101
Upper Address Bound: 192.168.248.254
On Windows 10 these issues seem to resolve automatically as Genymotion sets up the host-only adapter on its own.