Hello,
I can confirm that the 82-035 handle set and 82-315 key lock are compatible with each other but that does not necessarily guarantee that they will work for your repair. We would need to see photos of your original handle set and confirm if your hole spacing matches the 3-15/16" measurement of the 82-035 to confirm if these parts would be your best replacement option. Thank you.
We would still need to see photos of your original hardware to confirm but as long your dimensions match the ones shown on our product pages and your mortise lock is compatible with the new hardware, then you should be good to go.
You should only need to replace the parts that are broken, though you are welcome to replace all of them at once to reduce the chance of more parts breaking in the near future.
If the mortise lock and keeper are fine, there's no need to replace them. I only mentioned our recommendations for those parts because you mentioned that you may need to change the mortise as well.