Lets say I have a 6 letter word.
I send the 6 letters in two different ways. The first time I send it in the original order and the second time I send it in the following order: 4 5 6 1 2 3. There is exactly one removal in each observed set. For how many removals can I not correctly unscramble the message?
I did hand computation by checking each combination. I received 6/36 messages as not recoverable: (1,4) (2,5), (3,6), (4,1), (5,2), (6,3). Have I converged on the right calculations?
Your calculations are not correct. $(1,1)$ should not be decodable in the first case and $(1,2)$ should not be decodable in the second. These are just examples. You should still find that there are more cases you cannot decode in the first case because changing the order will mean more cases can be decoded because there are fewer pairs that are next to each other in both transmissions.