My understanding is that rear derailleurs are not that fussy between 8 and 9 speed. There might be some differences between the jockey wheels etc but not enough to make a significant difference.
My suggestion would be to try new cables first of all to try and cure the shifting problems. However, I’d be worried that on a cheap tandem frame you are causing it to flex leading to ghost shifts.
When you remove the cables you’ll be able to feel if the stiffness in the shifter is due to cable fricton or friction in the shifter itself.
Chain suck should be curable with a new chain and chainrings but watch out as you might also need a new cassette to go with the new chain.