In Settings → Notifications you can toggle each push type: video analysis results, daily tasks, direct messages, community chats, and re-engagement reminders.
Community rooms have their own per-room level (All / Mentions / Muted) set from inside the room.
Some things always notify you: money events, like a payment link being paid.
The red badge on the app icon matches your Inbox: unread messages across your conversations. Other notification types never change the badge. If pushes aren't arriving at all, check the iOS-level permission status shown at the top of the Notifications settings screen.