H. ergaster is thought to be just a later form/descendant of the African H. erectus, some don't consider it distinct enough to be its own species. Like I said, H. heidelbergenis evolved in Africa and later spread to Eurasia.

Well, yes, because we literally decide on what animals belong to our own genus. It's technically arbitrary, since we know Homo evolved from within Australopithecus. Homo habilis is generally considered the first species of Homo, though H. rudolfensis is about the same age. In our own line of descent it goes H. habilis, H. erectus, H. heidelbergenis, and then H. sapiens.

Homo erectus is thought to have left Africa first, about 2 mya, and persisted in Eurasia for quite a long time. Homo heidelbergenis left Africa later, maybe ~500 kya; the ones in Eurasia evolved into Neanderthals and Denisovans, while the ones in Africa evolved to Homo sapiens.

However, the existence of Homo floresiensis (often called Hobbits, because they're very short) in Indonesia adds a weird wrinkle to the question of "which species of Homo left Africa first?" Because it has a weird mix of traits that have led to very differing opinions on its classification. Some think it may be a direct descendant of H. habilis, some unknown early species of Homo, or even a derived member of Australopithecus, meaning one of those could have left Africa before even H. erectus, but we have no evidence for these ancestors in Eurasia, and most agree that it probably didn't evolve from H. erectus. This is actually one of the only really "missing links" left for our genus. It doesn't really affect our own lineage, but these are our cousins, so it's still an important question. It's kind of the weirdest outlier in human evolution that we know of.