You're right.
I've added a note on string ordering.

The actual tests contain digits as well as characters, and digits and characters do not have a natural ordering between each other (unless something like ASCII code order is assumed).

I think it can be closed now.

I will be happy to help you, but don't want to kill all the fun with this kata, not yet.
I did not test your solution, but it can be that it's not sorting what should be sorted.
And, in random tests, do you get errors, or timeouts?

I've tried my best ... but I have no idea what's the goal of this task.

I understand, that it's supposed to look like a riddle, but in my opinion Katas on this platform should be clear on what are the requirements and what's the goal.

I thought I've understood the task and passed the training tests, but then random test came and I have no idea what am I expected to do.

Please, provide a more clear explaination and/or an example.

That was a great journey through modular arithmetics. Thank you.

I did not think of it as a puzzle, rather an optimization of naive solution.
And you're right, it was hard to realize the goal. Hope it's a bit better now.

It is

nth ordered suffix of text engraved above each door

but yes, it's hard to see and also hard to realize what "ordered suffix" means.

Hey Elminster, without hitting the "train" button, it's not that clear what is expected.
Is this a puzzle? If so, add the appropriate flag.

It would be nice to have some random tests (to prevent hardcoding), and maybe shifting with very large values, or negative ones.

I didn't mean to change the tests, but the code snippet of initial solution.

C# version expects result as int[] in tests, but code snippet defines return value as List<int>.
It would be much better, especially for beginners, if the code snippet suits the tests.

Can you post some sketch of your code, or pseudocode of your solution, so I'll try to help. There is a lot of space for optimalization in this problem, but I'd like to know where to start.