Is it possible to create an MQTT API service for being consumed by others?
If yes, how can I do that and would that be of any use?
If no, why? Maybe it is possible to set up an API using some other IoT-specific protocol (like CoAP or something else)?
I thought that it would be useful to persist the sensors' data and the data generated by other IoT devices in some cloud for being processed and analyzed.
It already exists.
Look at things like Xively as a dashboard for MQTT data or the IBM's IoTF and Amazon's AWS IoT offerings