Looks like as good as you could get for ease of doing this!
You know where/how to turn off the water and let it drain? Ask before cutting , if not!!!
With a house this new, things are much more reliable and somewhat easier. My suggestions would pretty much match your idea of just punching a hole through the wall under the vanity to tie into the line. Decision for you to make is whether your incoming water will need to be from hot and cold or is cold okay for adding direct to the tank? Gets down to using what may be cold water which has to be added a bit more carefully so not to give them a chill or more work to add a hot and cold with a mixing valve setup to give you warmer water.
There are a couple ways to tap these lines. You might need to do some checking on what is handy for you to find but this would be my thoughts.
Take the compression fitting off at the pipe coming out of the floor, replace the current cutoff with a cutoof like this:
I would then use the existing copper to go up to the faucet or I might upgrade slightly to a braided flexible tube. I got to admit, I'm getting spoiled and don't like messing with copper tubing. But it will cost you $10! your decision. From the second outlet to the tank is much the same decision. Some type of tubing through the wall and a cutoff at the end to control it. For this tube there is a real nice (expensive?) braided tube that is great and will give you much comfort knowing it will not break. They can be had in like 6 foot length. They might be found around the icemaker supplies.
Second somewhat harder but also cheaper is cutting the supply tube above the cutoff and add a tee in the copper tubing. Cut the tube, put in the tee involves two compression fitting and easing the tubing back to fit the cutoff in. Not bad work if done carefully so the copper doesn't kink when moving it.
Draining to down below sounds like it's good.
Give it a bit of study and yell if it needs other ideas?