Warm nose in dogs indicates nothing but hard tummy and foamy vomiting can be due to canine bloat. For details, visit this link.
My German Shephard had this a while ago, we realised that it happened when she had eaten a rawhide on an empty stomach, sometimes when they eat stuff that doesn't settle it will create Bile which is the yellow foamy stuff. I gave mine some lighty scrambled eggs to settle her tummy and fed as normal when next feed was due, she was fine after a day. Hope this helps
Your dog might be having canine bloat which causes foamy/white vomiting along with shaking due to abdominal pain. For details, visit canine bloat.