If I spent ten days at a hotel (April 21st to May 1st) and paid for an additional 1 week (May 1 to May 7th) checkout on 8th before 11:00 a.m., but, I decided to pay for another week on the 7th, the day before checkout for the first additional week. Question: when is my checkout date? is it the 14th, or 15th (by 11:00 a.m.)?
It depends on your agreement with the hotel - you say specifically you paid for another week, so my calculations would make that the 15th because it prolongs your stay and you still have another night already paid, so its in addition to that.
But really, it depends on the agreement you have with the hotel - you can outright ask what day is your new checkout day for example. There is no world wide standard, and misunderstandings can happen, so be explicit and dont be afraid to ask.
Lambshaanxy’s comment and Moo’s answer seem to be on point. But, the wording of your question makes me wonder.
You stated, “If I spent ten days at a hotel (April 21st to May 1st) and paid for an additional 1 week (May 1 to May 7th)...”. Did you pay for ten nights accommodation (check-in on the 21st, check-out on the 1st)? If so, your purchase of seven additional nights twice means you paid for twenty-four nights. The math on this would obligate the hotel to provide you accommodations until check-out on the morning of the 15th of May, alternate accommodations until the 15th, or a refund for nights unused.
I ask this because I have extended stays at hotels many times. Regardless of if I prepaid the stay through an agency or website, or if I booked the stay without prepayment, the charge for the additional time (and original time in the case of no prepayment) was settled upon checkout.
This answer is coming from the perspective of a hotel located in the United States. Other countries may vary. Your best practice would be to ask the establishment directly. Though I have never run into a situation in any country where I have been so far that this did not apply. Voluntary nonuse of the accommodations without prior notification on your part would be a different story. Though, many if not most will allow early checkout without penalty.