Hi RRXing,
Thanks for reaching out to the Apple Support Communities. To get started, while there wouldn't be a way to turn off notification badges for specific conversations, you can turn off badges for Messages in Settings > Notifications > Messages, and ensure that the "Badges" tab is disabled.
Check out the following resource for more details on managing your notifications at this link here: Use notifications on your iPhone or iPad - Apple Support
As for leaving group conversations, is this a group iMessage or a group SMS/MMS conversation? We ask because you can only leave a group text if everyone is using iMessage. For more details, check out this next resource: Leave a group text message thread on your iPhone or iPad - Apple Support
How to leave a group text message
1. Tap the group text message that you want to leave.
2. Tap the group icons at the top of the thread.
3. Scroll down and tap Leave this Conversation. If you're on iOS 14 or earlier, tap the Info button
, then tap Leave this Conversation.
We hope that's able to help.
Have a great day!