When beginning, as you pull the tzitzit through the hole – do you have to keep the individual tzitzit strings in order so that the tekhelet strand is the first or 4th (blue, white, white, white – as in your diagram) or will any order do? And if, after you finish tying, the order of the tzizit in the hole changes – is that a problem?

Order within the hole is unimportant. There is an idea that the white string should enter the hole before blue one (this is a lone interpretation on the Gemara which states that one should start with white and end with white) – but this is certainly not something which would mitigate the fulfillment of the mitzvah (i.e, placement order is not maakev).

– Mois Navon.