Is it possible to only have a stutter or stammer in certain situations like when a child is really emotional?

Yes. Absolutely. And this is, in my opinion, the best kind of stammer/stutter to have. After all who, amongst us all, is fully comfortable in all situations? If a situation "causes" a condition, "training" can just as easily "cure" it. Practice makes perfect...
Between 2-5 years, . stuttering can occur in typically-developing children. If it lasts more than 6 months it should be evaluated. Anxiety makes stammering or stuttering worse. Don't pressure the child to " spit it out". In an older child Speech/Language Therapy is recommended. Evaluation by a child psychologist can determine if the child's symptoms fulfill criteria for an anxiety disorder.
Yes but . . . . You should consult a speech therapist preferrably one whose speicalty is stuttering. But yes emotion can make stuttering worse. At some times emotion may make stuttering better but mostly relaxation helps make it less. Still the client may have mild stuttering that is not easily observed at non emotional times.