Obedience training I've learned is a non stop 24/7 task. Especially with an untrained dog. Some dogs pick things up quicker than others. I didn't see if you mentioned you have her in classes with a qualified instructor.
My advice, get her into a class structure with a qualified instructor. Obedience training is not all about the dog. Obedience training teaches the handler what to do and what not to do. So your being trained as much as the dog is being trained.
I don't see a need for a prong. Persistance pays off in the end.
On the stay command. Try this. Put her in a sit. Hold up one finger and with authority command her to stay. Take a half step back for a second, then step back towards her and command "break". Treat and praise. Just keep doing it over and over. Repetition and persistance will pay off. Keep doing this and when she seems to be doing good at your distance, take another step further away. Key is, if she gets up or moves from position, you have to put her back into that original position. If she breaks for any reason without your release command, no treat. She will get.
With her rolling and licking from a down command seems like a submisive behavior. When she does that, it's not the desired action of the command. You need to put her back into a down position. Just like with the stay command, you have to treat at the proper time. Do not treat her when she rolls over. This will reinforce her to do this everytime you give her a down command. Instead, give her the down command, used with the hand signal palm down. As soon as she reaches the floor treat and praise. Keep doing that. If she rolls over from a down, no treat or praise.
These are things you will learn in classes. When to praise and treat and when not to. It is one of the most important things you will learn in dog training. Treating at the wrong times can make small problems harder to remedy. I was training my GSD on a leave it command one time in class. I was walking him by a toy he was not to touch. I was having a difficult time with it until my instructor watched us performing the task and informed me I was treating him at the wrong time. I was treating him just after we walked by the toy but he was still making eye contact with the toy. I should have been treating him as soon as his eye contact left the toy and was back on our walking. That one simple small minute thing made this training task a breeze. As soon as I corrected myself on when to treat, he picked right up on it and has been great with the command ever since. This transpires over to other training commands also. Thats why I mentioned getting into training classes are highly beneficial for both you and the dog.