Lithium batteries should not be bent, folded, spindled, mutilated, manipulated, shape shifted, or used in any process that involves changing the case's shape. The insides of these batteries are lined with metal plates (I forget what material specifically) that can separate, break and otherwise be damaged. Plates that aren't supposed to touch can also come in contact in cases of extreme deformation.
Will the battery still work? It may or may not, only way to know is to measure voltage or try it. Just be careful! There is a fire hazard with severely damaged lithium batteries, just youtube "lithium battery explosion"...but of course those are extreme cases and go long past the point of noticing "hrmmm, something wrong here".
The flex cable replacement depends on the severity of damage. Did you cut through any of the traces? Is this cable also severely "molded" to the point where it may have broken the traces?
My PERSONAL suggestion would be to just grab both if you can. It will save you time and frustration later! Good luck.