I can't speak to Code in your area, but when I built my house low voltage was not even something the inspector looked at. It's mostly common sense. You want to use cable that is rated for in-wall use - even better, use plenum-rated cable for everything. Cat 6 at least. And put a little distance between your low voltage and line voltage wiring, avoiding or at least minimizing parallel runs where possible. Minimize stud hole size, but leave room to run another wire in the future.
I would not terminate until the drywall and painting is done, rather install a low voltage ring or box, coil the wire inside, and let the drywall guy finish around it (I would put masking tape over it to keep dust and paint out). Also, I would suggest running at least 2 cables to each place you want ethernet because you will find uses for the extras - and it is easer to use multiple cables rather than install switches later on. Also leave a few extra feet of cable on each end for slack and future cuts/relocations.
And don't forget to label both ends of every cable with a unique number for each run. I used a P-Touch labeler and a numbering system of Cable Type - Room - Run Number, so 6-G-1 is a Cat 6 to the garage, run number 1, etc. I did run a few wires and left them hanging loosely inside the stud cavity only because I didn't know exactly what height I wanted to locate the termination within the stud cavity - but make sure it will be accessible and long enough to find it and grab it through your hole when you make it.
I also strongly suggest that you make a video every foot of every wall in every room after all line and low voltage wiring is done but before sheetrock. Trust me, you WILL use that video in the future to help you find existing wires and pipes and to locate cavities to run new wires that you didn't know you would need. Actually ran far more after-the-fact low voltage than I ran when I built the house - you will always need more wires than you think.