Welcome to the site.
Funny you should mention rabbit's. Labs took a turn away from strictly retrieving work many years ago and the use was rabbits. Some outdoor writter did an article in one of the magazines on rabbit hunting with "super charged labs". I would guess that any game that could be flushed would be fine. The only problem might be if the rabbit got out of gun range then came back around, your adult dog might be to close to shoot the rabbit. Haven't gone cottontail hunting in years but been doing a lot of thinking about it lately. With my last dog really old, I'm thinking of another dog and thinking more on the lines of a flusher. Getting to decrepete to chase a good pointer anymore and my horse died two years ago.
Is that a Savage mod 24 that little guy is holding? Got a good story about them if it is. I had one I got brand new in Germany about 1969, 22mag/20ga. Never used it a whole lot so some years down the road I traded it off on something up in Kalispell, Mont. About ten years after that I was in a second hand store in Greeley, Colo and there it was, I recognized the different front sight I'd put on it. Bought it and took it home and looked up the serial number in some old records and sure enough, there it was. Have no idea where it is now but I suppose it'll show up again someday.