Sounds to me like she may have stones/crystals...I wouldn't be satisfied with the answer the vet gave you...That's so out of character for her. The only way to tell is by x-ray or ultrasound....Remember that not all stones will show up on an x-ray. Miko produces ammonium urate stones which do not show up on x-rays, but as soon as they do an ultrasound, they can see a TON of urate stones...So, make sure they do both...
And the answer to your questions is yes, my yorkie has done this and the first time it happened, they told us that he didn't have a UTI, but they would send him home with antibiotics anyways?!? They were right, he didn't have a UTI, but he did have stones/crystals which they didn't even mention was a possibility.
I feel so bad for your little girl. Obviously something is going on...I'll be praying for her!!!