Is it worth $50- probably yes.
Is it worth investing a lot of money to make it into a first class riding machine- probably not.
Is it worth investing a lot of money in to restore it to like-new condition as a show-piece- probably not.
Any old ridable tandem will go for $50-$200 or so. So you definitely bought at the low end and in the right place at the right time could double your money. And if you just want to do an occasional around-the-block, that's the way to go.
As far as using it for serious riding, there's a lot of "ifs". Is it the right size? (My Huffy is on the smallish size) How does it feel? (my Huffy is pretty flexible) How's the weight? Is everything working, or do you have to start replacing wheels and hubs just to go around the block?
As far as cosmetic restoration, given that these things are not uncommon, you'd do well to be very picky about what you start with to minimize the work and maximize the results.
My guess on the sync issue- somebody had the chain off and didn't know the difference.
__________________
"be careful this rando stuff is addictive and dan's the 'pusher'."