I have a visa for a month with 15 days of maximum stay starting from September 5 until October 4. I will be traveling to a Schengen state on September 15 until 29. Am I allowed to stay until the end date of validity?
Generally, you are allowed to stay until October, 4th.
But keeping in mind, that your duration of stay may not exceed 15 days you are not allowed to stay until October, 4th, if you arrive earlier than September, 20th (September, 20th to October, 4th make 15 days).
So, if you arrive on September 15 (as you plan) - you are not allowed to stay until October 4. If you arrive on September 20 or later - you can remain until the end date of your visa.
I suggest that you have a single entry visa. If you have a multiple entry visa, you can leave the Schengen area and enter egain several days later staying until Ocotber 4. But even in this case the total duration of your visits may not exceed 15 days.
The starting day and the ending day both count fully, so from the 15. to the 30. counts as 16 days.
You need to come a day later, or leave a day earlier.
This can be verified by using the Schengen Calculator.