Humans can now turn into Super Saiyans because Goku made a wish on the Dragon Balls.A big part of the latter end of Z was Goku realizing he wouldn't be able to protect Earth forever and finding a successor who could take over when he was gone. Eventually he would have had to realize that just one person might not cut it and so summoned the dragon to give all humans the potential to become as strong as Saiyans so that anyone could become powerful enough to defend their world, should they so wish.
- Alternatively, Saiyan blood has simply spread so far throughout the human race that thousands of Humans now have Saiyan ancestry. Saiyans do remain in their prime longer (apparently to enjoy fighting longer, Nappa had to be pushing 50+ in the Saiyan Saga and didn't look any older than his late 20s, early 30s), and this could also mean they have longer life spans than humans. Thus, Goku and Vegeta could have had at least one additional partner after Chi-Chi and Bulma died, same with Gohan, Goten, and Trunks.
There's more than one set of Dragon Balls because the ways of the dragon clan have spread.In the original series, even a child like Dende is able to create a brand new set of dragon balls, provided he had the commitment to perform a 100-day ritual. The Dragon Clan is now a basic playable class; it makes sense that several of the now thousands of namekians on Earth in that school would have made their own sets before the events of the game.
- Notably, EVERY surviving Namekian is a member of the Dragon Clan, because all of them (except Kami and Piccolo, who are themselves Dragon Clan) are Guru's descendants.
Sometime in the mid-800's, Pan becomes known as Grand Master Pan.She's a world renowned martial arts instructor by age 15 and doubtless begins competing in and winning Martial Arts Tournaments. In time, she becomes a master of every known martial arts style. Once she has done so, she begins referring to herself as Grand Master Pan, and the rest of the world concurs.