Full House/WMG

Joey is the biological father of the Tanner girls.
Think about it: why else would a grown man with a successful career live in his friend’s basement for eight years? Jesse living there for so long makes a kind of sense: the girls are his only living link to a beloved sister and later on his wife provides them with a badly needed female authority figure. But Joey? What does Joey bring to the table? He initially moved in because the suddenly widowed Danny needed help raising three young girls on his own and offered the then-struggling comedian free lodging in exchange for around-the-clock babysitting. Sure, that makes enough sense for a sitcom. But after awhile, DJ and Steph are old enough that they don’t need a babysitter anymore and are often seen looking after Michelle, Nicky and Alex. They don’t need Joey as a source of day-to-day emotional support, either. They are well adjusted young women and turn to Aunt Becky for advice way more than any of the guys. So why does Joey insist on staying with the Tanners? And why does Danny allow it? It’s because both men know the unspoken truth: that Danny is infertile and that those very blonde Greek girls are either the product of a three-time sperm donation or a decades-long extramarital affair.
 * Come to think of it, Nicky and Alex are mighty blonde, too. And Jesse was on the road an awful lot around the time of their conception...
 * Or maybe they all just got blonde hair because of their mother, who is also a blonde.
 * Maybe Nick wasn't Pam's biological father.

Jesse's last name in the first season was just a stage name.
In the first season, Uncle Jesse's last name is Cochran. In the rest of the series it's Katsopolis. This could be a subtle reference to The Beatles where Ringo Starr's real name was Richard Starky or other bands whose members had stage names. Danny even mentions in the first season Jesse's band is Jesse Cochran and the Rippers, while the rest of the series has it as Jesse and the Rippers.