Replacing my sliding door handle and lock set

Sure, I think we can help with these parts. Start by looking at our 82-089 handle set and 82-265 mortise lock. I think these may be the parts you need, but review their specs below and see how well they compare.

Dimensons for 82-089

Dimensions for 82-089

Alternate image for 82-265

I think your mortise looks different here because the face plate fell off. Your first batch of photos showed a white face plate that covered it and allowed it to mount to the door, but that appears to be missing here.

What you're showing here is closer to the 82-239. Granted, this is identical to our 82-265 except that it doesn't have a faceplate, so if you wanted to play it safe you could just use the 82-239 and, if a face plate ends up being necessary, you can just reuse your original.

Sure, I didn't know you wanted a key lock for this. The 82-089 doesn't come with a key lock but our 82-086 does. This should work perfectly fine with your hardware.