1. Yes, your annual HSA contribution limit is the limit for your HDHP policy ($7750 Family) time 6 months divided by 12 ,(so, $3,875) assuming that your spouse who owned the HSA was under 55.
2. Yes, again. You could contribute to the HSA before you had the FSA coverage, but note that that the limit is on an annual basis. If you contributed the whole $3,875 on July 1st, TurboTax would be none the wiser, and not report the excess.
3. Yes. Understand that you don't make any contributions to the FSA (your employer does), so there is nothing for you to refund here.
I see that I was answering another question than what you asked. Opus was right to catch that you did not contribute enough to cause an HSA excess in 2023.
"But later on Turbo Tax mentioned there were errors on Tax Form 8889-T and it asked to provide a checkbox for each month in 2023 that had HDHP plan." Please tell us exactly what the error said, because TurboTax can become confused in the Review abut your HDHP coverage, but what you seem to have entered should not have caused it.
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