I recently tried deleting an object in the model tree which affected the proceeding constraints. Is there a way to restore the original constraints after unfixing the part locations or does this require re-constraining every part?
If the constraints are directly reference the deleted part, you will not be able to recover them without reconstraining every part. You can use the "edit references" under the RMB on each part to quickly re-select new references. Also, if some of the failed components are children of other failed parts, they will correct themselves when their parent is re-constrained.
FYI, Edit References is the chain looking icon, it's also under the model tab, operations drop down arrow.
I would also suggest using suspend under Children Handling, especially if you are going to repair the model as your next step. You will not have to delete a fix constraint.
Another good suggestion, if you are replacing a part, is to use the replace command and use the Edit Ref Table button to define reference replacements.