If the connection light is on, it means your gateway has gotten an IP address and is connected to the network.

I don't believe there is mechanism to check IFTTT messages as they are just HTTP requests made to an IFTTT endpoint.

As for MQTT, you could run your mosquitto server in a verbose mode and see if the gateway establishes a connection with it or not. You should see the gateway connect as a client as long as it can find the broker successfully. Another things to try would be to use a MQTT client such as MQTT.fx to make sure your broker is indeed healthy.