newborns have this sort of bumps.. as you know they are "new".. so anything can develop a rash.. new clothes, soap, food.. etc..

I think if you expose them to everything.. with measure they can build antibodies and become immune... some things might not be possible (as allergic persons still exist I´m one.. allergic to egg).. but one can control them by knowing what´s causing them..

in the case of the doggies.. it can be the dander.. which is what makes peole allergic.. so bathing and good ventilation.. there are also airpurifiers you can try..

but first don´t panic.. it might just be a simple baby reaction to anything new..