Your phrasing and terminology are very confusing. So I am guessing on your questions.
during the-period-after-the-surgery – The way you had it written, I see a conflict between “During” and “After”, unless there is something missing you didn’t tell us.
I could say ” During the one-week trial period after the purchase of this product, you can return the product any time and receive a full refund, no question ask”. I am using the same construct but with complete context.
I think the main "bone" of the phrase is "during the surgery." – What is a “bone”? Do you mean “core”?
2. Are (Do) they mean the same thing? No, if you refer to “during” and “after”.
They would feel right, (wouldn’t it? ) isn't it?
They would not feel right, (would it?) is it?
3. In the sentence, I feel the use of an indefinite article "a" consecutively and repeatedly do not render a pretty picture as to the sentential construction is concerned. What do you think? --- What sentence? Please clarify your question.
A Quaker is a ("the" is better???) person who belongs to a Christian group called the Society of Friends. The answer is “no”.
A true friend is someone who will stand by your side in time of trouble. [ Not “the”]
The only friend who stood by me what Paul when I was feeling down. [ Not “A”]