Exam exceptions for student with disabilities
Exceptions can always be made under certain circumstances. Interestingly in your long and detailed question, there is no trace of a reason for requesting a delay other than your general condition, which I cannot see as a reason by itself. The university/department probably has some rules or policies for such cases. There are usually ways for people with different disabilities to take an exam in parallel with the scheduled exam in a a separate room, sometimes with specific tools and help. I can for example see that hospitalization and family funeral can be clear cases for such exceptions. 

So if an exception is made, what would you have to expect? To provide exactly the same exam twice seems extremely unlikely for the reasons you provide. So from the teacher perspective, making a new exam is not something to take lightly, it requires a lot of work (if done seriously). In other words, you would get a different exam designed to be of equal difficulty. From a student perspective a new exam can always appear either easier or more difficult based on what and how well you have studied but in general, the exam would not intentionally be made more difficult.