I was thinking about that, too. Her growl is definitely different from the growl she had when she was eating. Still, I still think discouraging her growling would be better because of her food aggression problem? What do you think?
My little Jose, a Chihuahua mix growls like crazy when we play tug, he shakes his whole head and body and really gets into it. He wags his tail, sticks his rear up in the air. I mean he has a wee of a time. I play growl right along with him. It's play and it's a healthy outlet. Sometimes he gets to "win." Sometimes I do. I say, "give" and he lets go. He "talks" to me too. He stares at me, sticks his rear up, wags, pounces and all the while I encourage this funny and cute "talking" he does. It's a combination of a growl and this human like, breathy, long drawn out vowell sounds. It's hilarious. He's playing. In fact, I'm trying to put it on cue...."talk to me." I use a very playful voice. So, do I think you should discourage growling? No way. Not in play and not if he's using it as a warning to stay away from his food. Never discourage a growl or punish it. It is communication. Don't sever the lines of communication.
Now the feelings he has about his food, this natural instinct to defend it can be modified. Never mind about the behavior or signs of his defending it. Let's get to the core problem of his worrying about you taking his food.
Hand feed for a while. Then putting handfuls of food in the bowl, having him sit/wait first and then putting the bowl on the floor. Or better yet, before you use the floor, hold the bowl for him for a few days then use the floor where you set it down. When he finishes that handful, give him a piece of cheese or hotdog while you pick up the bowl. Put another handful in it, asking for a sit/wait or down/wait (whatever) while you do this and while you set it down. (If he breaks the wait, pick it up fast, before he gets to it) Teach him a release word like "OK" which means he can leave his sit/wait and have his food. Do this handful by handful. When you take his bowl up, trade him for something extra special.
Later, after you do this for say a week or whatever it takes for him to look comfortable with this procedure, try putting the bowl on the floor with part of his ration in it. Walk around the kitchen a little, come by his bowl and drop an extra special yummy thing in (hot dog, cheese, last night's chicken, whatever) Do that for a few days. When he looks relaxed to you with your walking near his bowl, try bending down and letting him see the hunk of fresh meat in your hand, put that in his bowl. In other words, you're conditioning him to see you as his provider of resources. That's what makes him trust you and be comfortable with coming near his food.
After you've done this for a while and he isn't shifty eyed or growling or nervous/tense looking, when he doesn't pause while he's eating, stop chewing, look tense....when he just keeps eating and doesn't pay much attention to your presence, you can be pretty sure he's feeling relaxed about you coming near his food. When you can walk around doing your thing, walk past him, drop something in his bowl, you can start feeding him his whole meal in the bowl, asking for a sit/wait/OK before he goes for it. If he breaks the wait, pick the bowl up first before he gets to it. Replace him in a sit/wait. Teach him that only by sitting/waiting polietly and waiting for his release word, will the bowl remain on the floor. Do it as many times as it takes.
Now he's to the point where he's getting his whole meal in the bowl. He sits/waits/OK and he gets to go to it and eat. ONE time during his meal (let's stop harrassing him at this point) hand feed him a piece of meat or cheese and pick the bowl up, move it over a couple of feet and he can eat again. Show him that when you handle his bowl, not only does he get a super yummy treat, but he ends up getting his kibble back too. LOL. Then leave him to eat in peace.
Be all pleasant and relaxed through all of this. Do not be tense or nervous. Take it in stride, like this is no big deal at all. Don't let him sense any nervousness or worry in you. Be confident and matter of fact and praise him for good behavior.
One final thing: Don't get all bogged down with all of this. I mean read this stuff, get the gist of it, the point of it, why you're doing these things and then go gradually to condition her. It doesn't matter if one little thing is left out or not done exactly as written. This is not a test. LOL. The point is to get your dog comfortable with your coming around her food and for you to be relatively safe in doing so.....and to go slowly.
This procedure, if it is taken gradually....probably over the next two or three weeks will make your dog more trusting and comfortable with you coming around the food, will make her see you as her provider of good things and not one who is going to take away her food. This will make the warning growl go away. She'll stop growling at you because there will be nothing to worry about.
Just remember...
never ever punish a growl. It teaches a dog not to growl. A growl is a good thing. It
communicates to you to stop doing something. It gives you time then to work around it, counter condition whatever problem the dog is having, like we've been talking about. When you supress a growl, say "NO!" or any other punishment, the dog is likely to bite without warning. YOu don't want this, right? In play, a growl is not a warning, unless your dog is not playing. But you should be able to learn to observe their body language and expressions and know. Look online for something which shows you body language. I don't know how to explain it because I've just grown up with dogs for 47 years. It's like any language. When you grow up with it from a very young child, you just know.
I keep mixing up he and she. OK...so this is a she. Good heavens!
Oh, your question about running. If you think she's just having fun and chasing, that's fine. If she is getting too riled up and getting into that prey drive thing, sinking her teeth into you or snapping, no, don't let her chase you. I would never let her chase little kids. They see kids differently sometimes. As long as she keeps her teeth off of people and doesn't get carried away, it's fine. Teaching a retrieve is a good thing. You can play fetch with her, frisbee too.