Not sure what's the fuss but here's my take on the whole thing...."Does D just naturally take more breath to play?"

Only if you play in a "more air" manner.

"How can I get it to piano or pianissimo so the transition between C and D is smooth?"

As mentioned, the correct overtone (blowing too low or squealing too high) is basically determined by the speed of the air and not the size of the air stream. Beyond a certain point, the size of the airstream gets too large. You would be wasting air and this may sound breathy or airy. The tone may also be very unfocused and you will not be able to play higher notes softly. Long tones may help of you know and understand how to work with your lips and breath support but this may require a good teacher that understands these things and how you are applying yourself.

I say that many flutists use to much air. That beginners tend to think that they should blow a lot of air and this teaches them not to learn to use their lip muscles to control the tone quality. ALso, some teachers think that the flute needs to be approached from getting a loud sound and then learn to play softer. Maybe it should be the other way around. Learn to play soft and clear and the loud sound will be easy.

So an exercise to be done without using the tongue:
1)Inhale.
2)Close lips.
3)begin to blow and allow the air to come from one small round hole in your lips. Focus, adjust, listen, pay very close attention.
Do this on all of the fingerings that you have learned on your flute.
If you can do this with the least amount of breath support and lip tension while getting a full clear tone, you may begin have something.

Just one approach ....for sure.

is the funny looking key the G# key?