If I delete someone's post on my Facebook page does it delete the comments on that post too?
Yes it deletes all comments, likes and shares of that post. I answered and tested this before in 2012, I'm sure it still applies now
I posted a status update to a test user Betty
Then had three other test users share it one after the other
So Tom shared it > Joe shared what Tom shared > Patricia shared what Joe shared
Then for completeness I had Betty share it again
On delete, all posts are gone from all test users, including the re-share Betty made.
As long as the user shares via the action link all subsequent shares will be deleted if the original is deleted.
But if the user manually copies it or saves a photo then re-uploads then no.