I do them in my living room and they work great. Really all I've needed is a bench (my coffee table), a yoga mat, a medicine ball, and a cheapie Amazon set of weights (https://www.amazon.com/BalanceFrom-GoFit-All-Purpose-Dumbbells/dp/B076XHDM7K). You totally don't need to join a gym, though if you can bum a friend's Planet Fitness card, go for it.
YES! I started having knee pain late into BBG 1.0 (week 9 or so). It turns out I tore my MCL (inner knee) and my knees felt like they were on fire some days but achy others. I made sure to rest them when I could and wear knee sleeves on BBG Leg days. I bought some simple compression knee sleeves (https://www.amazon.com/gp/product/B019ZK0WS0/ref=oh_aui_search_detailpage?ie=UTF8&psc=1). Most importantly, I made sure to be more aware of my side-to-side movements. I keep my knees straight when doing the jumping moves and make sure to stretch really well.