To do things halfway means to leave the job or task unfinished, to fail to do much that was necessary or expected. Compare "to take half measures".
You could say of a person who threw a party, say:
It was quite a bash, with live music, lots of food, and plenty to drink. Dmytro doesn't do things halfway when throwing a party.
The speaker in your video is then deriving a new phrase from that familiar locution, patterned after another locution. See this question. Only here, it would be do + preposition or do + adverb (depending on how you analyze "halfway").
Nature doesn't do halfway is like saying of a guy who always drives his car too fast:
He doesn't do "slow".
And with means "with respect to, regarding, involving".
With regard to blue (creatures), it is not in Nature's nature to do anything other than a bang-up job.