The Apex does not support SNMP. Yes, you can set static IP settings regardless of whether you connect via Cat5 cable or WiFi. Yes, it does reply to pings.
Heartbeat is useful, but not timely enough to take immediate action, and heartbeat cannot distinguish between a power failure or a basic network outage.
The best solution is to integrate the Apex with a UPS battery. It can then detect when it’s running on battery power, and as long as your Internet router is also on UPS then the Apex can send you an alert about the power outage. It can also be programmed to shut down high-power devices during the power failure to prolong the battery life. I wrote detailed instructions to set this up in my Apex Power Monitoring tutorial:
https://www.reef2reef.com/ams/neptune-apex-programming-tutorials-part-4-power-monitoring.698/