When does my membership year begin? On the 1st or the 3rd of the month
 

As the title implies.

My membership year runs from January to December, i.e the same as a calendar year. In 2019 my Elite Plus requalification period resets, and I have a flight booked at 10 minutes past midnight on the 1st of January. Normally the miles reset on the 3rd of January, so I'm a little worried that the miles credited on the 1st of January won't count towards my requalification, or are the miles backdated to the correct date? It would be a shame to miss out on the 9000 status miles.

Your new period starts on January 1st,since your membership year runs until December. Does not matter when the miles will be reset in your account. I re-qualified in first membership year and account page showed 100% done, when the second membership year started it shows 73% done. Btw. I used my upgrade vouchers on the first day of the new membership year, so they are posted to there automatically. You would not miss out your 9000 miles unless the flight would be changed to 23:55 or earlier on December 31st.

Based on my own recent experience (end of membership year 31st of March), the flight taken on 1 April is posted "above" the renewal line in the account. However, in the mileage statement, the miles of this particular flight are reflected as part of my total first year status renewal balance. Now I'm in the fortunate situation that I will easily pass the 40000 threshold even without that particular flight ...

I had a quick look at my mileage statement for July 2018, and I noticed that the flight that I took on January 2nd of 2018 counted as my 1st membership year mileage, despite my membership year beginning on January 1st. This has me somewhat worried that those 9000 miles won't count towards my new membership period. Can this be corrected retroactively, or should I simply ask the check-in agent to remove my FFP from the boarding pass, and have it retro claimed after the flight?

I achieved Elite on 14th September 2015 and renewed it with flights until 30th September 2016.

So in your case, it doenst matter when or how you claim for crediting the miles; it will count on the date you fly.

Since your E+-Status (not membership period!) started on 2nd Jan 2018, your miles for 1st Jan 2019 shall be counted.

My Elite Plus membership was earned on December 10th 2016, so my first membership period runs from December 10th 2016 to December 31st 2018. In other words I expect my next membership period to begin on January 1st 2019. However, whenever my membership period resets it doesn't reset on the 1st of the month, but rather the 3rd (my previous Elite status ran from February 1st), hence my concern that the flights taken will not count towards my new re-qualification period.

Why should it reset on 3rd of Jan? I dont understand.

The TK system runs its status sweeps on the 3rd of the month. That's when the renewal is processed and upgrade vouchers get credited.

I think the idea of leaving your FFN off and retrocrediting does lower your risk, although I'd note that most of my miles don't credit within 2 days of departure anyway..

My upgrade vouchers were credited on 1st of the month, although that was for second year not the new membership period renewal.