Some strange behavior on the F5 when one pool member is down

Can you post the config of your health monitor? I would guess the problem is the monitor is incorrectly configured. You should have a receive string configured in the monitor (something like 200 OK) if you want to mark the pool member down based on a 404 response. If the receive string is blank, any server response will result in the pool member being marked up, essentially the same as a tcp monitor.