Yes. The detergent in toothpaste-sodium laureth sulfate (sls) can cause soreness and even mouth ulcers. There are sls free toothpastes available. If persistent problems see doctor.
Possibly, but... From your description i would say that it may be your toothpaste, but it also could be something else. The fact that your gums are peeling, i really would recommend seeing a dentist quickly for an exam and diagnosis as to what the problem really is!
Rule out infection. Your throat pain could be caused by an infection. Seek medical consultation to rule out that possibility. Peeling gums may have multiple causes. If you have recently changed toothpastes and/or are using a tartar control toothpaste try a different toothpaste for a few days to determine if the gum irritation goes away. The toothpaste may be causing the irritation.