Welcome to Microsoft Q&A, and thank you for reaching out.
Based on your description, from my research, unfortunately both request is not supported for shared mailboxes.
First, regarding preventing user from deleting mail in Shared mailbox, it is not supported according to Microsoft documentation:
If your goal is to reduce the risk of emails being accidentally deleted, Microsoft recommends using a Microsoft 365 Group instead of a shared mailbox, as it provides a different mail-handling model that helps avoid this scenario.
You may also work with your IT team to configure retention policies. This allows administrators to recover deleted messages through eDiscovery if emails are removed by mistake.
For the read/unread status, this is also not supported on a per-user basis for shared mailboxes. Exchange stores the read state as a property of the message itself, rather than as a per-user attribute, so any change applies globally.
For a workaround, you can consider encourage team members to use message reactions (e.g., thumbs-up) to indicate they've seen a message, this provides lightweight tracking without altering read status.
I hope this helps.
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