Automatically Close Messaging Session or Change Agent's Status

Use Case: We have a use case where agent always initiates a Text messaging session asking for photos (MMS) of the installed product through an Agent Initiated text. The customer responds with the images and those images get assigned to the case object associated with the Messaging Session. Agent who had initiated the session is then supposed to accept incoming response from customer through Omni-Channel and then "End" the session to properly close that interaction.

Issue: sometimes an agent forgets to accept the session and closes the Messaging tab and goes offline in Omni-channel. That causes the response from the customer to the specific agent to go back to the Omni-channel queue and it then gets assigned to a different agent who is NOT expecting the response from previous agent's customer and will not accept it.

This behavior sometimes spirals into a snowball causing a lot waiting messaging sessions in the queue and we eventually we have to go in the list view to close these waiting messaging sessions manually.

Expectation: We would ideally want to automatically accept and close certain messaging sessions with no action required from the agent. And then make the agent offline if he remains online for a certain period of time inadvertently.

Question:

1. Is there a better way to handle this use case where the photos sent through a particular messaging session get automatically attached to the associated case and the session is closed without any further action from agent after they have sent initial photo request?
2. Can we have messaging sessions always go to the specific agent who initiated it and if that agent goes offline for some reason, the session goes back to New/Waiting status until that agent comes back online?

Please let me know in case any additional details are required. Appreciate everyone's input. #DSOfficeHours