1. No
We very consciously did not put timestamps into the MQTT protocol, mainly because of the notoriously difficult problem of distributing accurate time to edge devices.
If your device knows the time, and wants to put it in a payload, that’s up to the application, and the receiving applications will know how to extract it, of course.
MQTT v5 has the notion of message time to live - that’s based on how long it sits in a queue in a broker, and does not rely on knowing wall-clock time.
I hope that helps.