HSA and Medicare or Social Security application vs enrollment

I turn 70 in March. I am currently employed, covered by HDHP, and have an HSA. I plan to start Social Security at maximum, first payment in April.

I thought that this would mean I would need to stop HSA contributions April minus 6 = October. So I maxed out my HSA contribution in September. 

Social Security recommends applying for benefits 4 months before start, so I began my application in December.

 

1) Today I was informed by Social Security representative that this would mean I also applied for Medicare in December, and my enrollment would then be Dec-6 = June! Is this correct? Does Medicare enrollment begin six months before the initial Social Security application regardless of when SSA benefits start, assuming the applicant is over 65?

 

I apparently can cancel this application, completely undoing this issue.

 

2) If I apply in late March, with Social Security benefits to begin in April, will my Medicare enrollment then stretch back in September? To have enrollment begin in October, would I need to apply for Social Security benefits in April?

 

As of December 31 2020 I was NOT on Medicare, and was covered by and paying premiums for HDHP for the entire year. I therefore believed that I could contribute my full annual allotment for HSA early in the year ( completed in September), and there would be no conflict. Nobody I spoke to at that time at my employer, Medicare, or SSA indicated anything different.

3) But now it seems that even if I start my SSA application after April 15 2021, if Medicare enrollment gets backdated into October 2020, then my HSA maximum gets reduced by 3 months, and I've over-contributed by those 3 months, by 25%. Is that correct? If so, I need to withdraw that 25% and any interest/income earned by that 25%, before April 15. Is this correct?