There’s a sprawling family tree on the wall of the new Barrymore Film Center. In a red box at the center: John Barrymore, the vaunted actor of stage and screen. Born in 1882, he would become one of the pillars of America’s first Hollywood — Fort Lee. In a golden box off to the left: John’s granddaughter Drew Barrymore. The actor, born in 1975, is now a talk show host and social media sensation known for taking great joy in the little things, like running into the rain. The $16 million Barrymore Film Center, a new 21,500-square-foot cultural space in Fort Lee with a 260-seat movie theater and m…