[How] can I find out through a command whether DHCP is enabled?
Provided that adb is setup in PC, execute:
adb shell dumpsys wifi
In the output locate the string Configured networks. Following that would be the info on all the saved Wi-Fi networks in your device. For each network, inter alia you would see a line with the string IP assignment. Based on your configuration for that netowrk, it would either show STATIC or DHCP.
Note that you must have root access if you intend to run dumpsys wifi on a device through a command-line app since ordinary apps are usually not privileged to access a dump of WifiService.
Here's the relevant screenshot captured for Android 4.2.1:
The findings are valid for stock Android 6.0.1 as well.
Aforesaid information however doesn't query whether the router itself is capable of running a DHCP server.