Hi @gagan.shah ,
to 1. The second c8y_connection is only “green” when subscribed for operations via REST API (long-polling) or you have a MQTT client connected which is subscribed to “s/ds”. See also here: Device management - Cumulocity IoT Guides
to 2: Use RestTemplate for the empty PUT request. Here are some nice example of Alex. You can see how to set the headers (authentication etc.), just adapt it to inventory API and you are fine.