The Big 10 has the highest media rights contract

Penn State is the only team in the Big Ten+6 without a protected rival. 

Historically, Penn State's rivals were Pittsburgh and Syracuse, but they have not played either since 2019.  Penn State's third rival is West Virginia, who they did not play against for 31 years, until playing again this year.

Penn State's true conference rival is Maryland.  Penn State pushed hard to add Maryland to the Big Ten in 2014 to give themselves a true rival.  The two had not played in 21 years, since Penn State's second year in the Big Ten.

Navy and Temple are the next two biggest rivals, but Penn State has not played either since 2016.

Penn State's next-biggest rivals are Ohio State, Michigan State, and Rutgers.  They have played each every year since the beginning of the Big Ten+4, along with lesser rivals Indiana and Michigan.  Penn State has played Ohio State every year, since the beginning of the Big Ten+1, which, I think, was part of the agreement to get Penn State onboard in the first place.

Michigan State and Penn State play for the Land Grant Trophy.  They have played every year since 1993, except for the three Leaders/Legends years.  

Minnesota and Penn State play for the Governor's Victory Bell.  Minnesota played Penn State in Penn State's first Big Ten game back in 1993.  The rivalry is partially protected with Minnesota and Penn State playing more often than normal Cross-Division opponents.

As for the nickname, Penn State lies at the foot of Mount Nittany.  The Nittany Lions are mountain lions, who historically lived on Mount Nittany.  The Nittany Lions were chosen as a team name to counter Penn State's rival at the time, Princeton, and their Bengal tiger.  By the time that the Nittany Lions were chosen as a team name, actual Nittany Lions had been extinct for at least a decade.