1) Years ago, I bought a used Huffy tandem single-speed. My daughter and I rode it around the block. It seemed to me like the frame was flexing, ie, twisting between us. Maybe that was my imagination. Maybe not. My daughter was maybe 120 lbs, not exactly a heavyweight. That bike had a step-through frame in back, had small tubes. I would expect a tandem with bigger tubes not to do that. My current CoMotion doesn't noticeably do that, but has much larger tubes and rear is not a step-through frame. So answer is "maybe".
2) Probably not. If it seems pretty decent all around and there's just one problem, sure, fix that problem. If it seems like a problem all the way around, do something different.
3) That could be cheap crap or that could be a problem for a bike shop to work on. And they may not do much better, or may not be interested in fooling with it if it's a lost cause.
On my $100 mountain bike (single bike), I could adjust derailleurs all day, but I either had to live without high gear or without low gear on the rear, it just wouldn't reliably go the full range. On that bike, I also had crummy brakes, so between shifting issues and brake issues, I was kind of glad when it got stolen. My single-speed Worksman cruiser was an upgrade from it.
"Compatible" on the tandem means you ride at similar cadence (or you'll drive each other nuts), that you're willing to put up with the arrangement, and not a whole lot else. So most of the female cyclists I know that are around my age (54) have ridden a tandem at some point and were okay with it, even if it wasn't their preference. One that wasn't, her husband said they rode to the end of the block and she said "No way", that was all it took. My current stoker won't even consider riding a tandem with her boyfriend, he's a spinner and she's a masher, and it's pretty obvious watching them ride single bikes that it wouldn't be workable on a tandem. For me, as captain, it took about 1,000 miles before it didn't feel "weird" or "wrong", but still felt okay, if that makes sense. Anyway, if you ride it around some and you're comfortable with the idea except that the bike is crappy, go for a better bike.
Sometimes, the larger bike stores will rent tandems, you might check into that also.
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