When someon is getting a skin graft, does it have to be their skin?

Yes. If you use skin from somebody else, the body will gradually reject it (unless it's from your identical twin). That being said, temporary coverage with skin from a skin bank is frequently used - it's just not a permanent answer.
Yes, at least part. Although combinations of grafts (xenografts) have been used, the best results are obtained when at least a portion, especially the keratinocytes come from the patient. The dermis may be obtained from other sources.