What do you think about the specifications that the project requires, should I stick to your code or should I add something from my own code?
I would stick to my code, your alarm isn't going to work properly due to its comparisons as I mentioned in my previous comments. But if you want to improve the code I modified, you can change the adjust_interval_up and adjust_interval_down buttons to be synchronized to their own states rather than the clock (make their own process with their signals added to the signal sensitivity list and add an extra asynchronous condition to zero the counter on the original process). If you don't make a change like this your alarm is going to take up to an hour to adjust its timer range.
Does your simulation correspond to a time of 1 hour and should there be alarming on the simulation?
Yes, if you have a 1/60 Hertz clock signal. And you must have alarming on the simulation as it is crucial to show that it works.