a dog bite is considered dangerous if we do not know the vaccination
status of the dog. but here in your case as it is a pet dog of your neighbour it should be less worrisome.
now coming to your qs. of having developed rashes. it is most likely to be allergic. whether it is allergy
to the saliva or allergy to the injections can be found out with detailed history and examination.
also the benefit of injection Is much more than the allergy if controllable.
hope it helps. should bring this into notice of your doctor who's vaccinating and seeing your child.