"Since I contacted the broker/administrators on excess contributions and initiated paperwork before the tax deadline; can I just handle the case 'as if the funds are already refunded'??"
Yes, it's the date that you officially requested the withdrawal that matters - you can't be held responsible for the amount of time the HSA custodian takes to actually refund the money.
"Does filing tax extension gets me more time (till Oct 15 2021) to get these excess funds refunded? "
Yes, for withdrawing the HSA excess contributions, you have until the extended due date or 6 months after the original due date. See the form 8889 Instructions.
As for the 401(k) return of excess contributions, I am afraid you are out of luck. The basic IRS statement is "Additionally, the corrective distribution must be made be made no later than April 15th following the close of the calendar year during which the excess deferral was made. See IRC Section 402(g)(2)(A)(ii). For example, excess deferrals made during 2016 must be distributed by April 15, 2017. This April 15th deadline is not postponed by extending the filing of the employee's federal income tax return."
Worse, for 2021, the IRS declined to follow precedent and move the date to May 17th, to follow the new tax deadline. "If an individual defers more than this limit for 2020, the excess deferral amount plus earnings must be distributed by April 15, 2021. Excess salary deferrals not withdrawn by April 15 are taxable in 2020 and again when withdrawn. The date to remove excess salary deferrals has not been extended." See under April 8 at this IRS News Page.
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