Whenever the cord is plugged into my Beats (can't use wireless because charging port is broken) audio won't play. If I...

It is possible, but you need reballing equipment and a precise knack for soldering

I'm pretty sure the charging port is broken. I have the same issue on my Beats and that's where all the signs seem to point to. I advise not paying Apple to do it (bc it's expensive af) and buying one and doing it yourself. I'll update when I get the chance to replace mine.