Hello ihsangoren,
Ensuring the Mail app is working as expected on your Mac is important, and you’ve come to a wonderful place for help! It sounds as though emails aren’t marking as read after opening and reading, and not deleting as expected. To confirm, the emails that aren't marking as read, are there multiple emails in that thread? If so, have you verified that you're scrolling and viewing each unread email in the thread? If not, make sure each unread email is read.
Regarding the emails showing deleted after closing and reopening Mail, have you tried restarting your Mac? If not, restarting can be helpful in reaching a resolution. To restart the Mac please click the Apple logo towards the top-left, followed by Restart in the menu.
You also mentioned this issue started after updating to macOS Monterey. There have been several updates since the initial macOS Monterey release. Have you confirmed you’re running the latest macOS version? If not, you can find the steps for updating macOS in the article below.
Update macOS on Mac
Take care!