It's not uncommon to get frustrated with a dog when you haven't been provided with the tools you need to set boundaries and be a leader, instead of simply trading for items and trying to use "Positive Reinforcement Only" training methods. Some leadership skills might help.
It wasn't the fact that you pushed the dog away from the bowl, it's that you were frustrated when you did it. A dog can sense this and you could end up being viewed as unpredictable, which can cause the dog to lose trust in you.
Don't beat yourself up over it, just move on and don't do it again. I know more than a few people who have started out trying to impliment training without leadership, and also got frustrated and did things they shouldn't have, or regretted. A Pack Leader must also be a trusted and consistant leader.
Since your pup is now biting, I would get a behaviorist who understands dog psychology, can teach you some leadership skills, and help you and your dog within your home. No need to rehome the dog if you're willing to stick with setting some boundaries and keep your temper in check.
In the meantime, reading Cesar's Way by Cesar Millan and sticking with his principles by putting the needs of your dog *FIRST* could help you understand how your interactions with your dog are being "read" by the dog and give you some basic leadership principles to help you.
I highly recommend this book.
Good luck and don't give up!