you have a lot of questions, i'll try to answer a few.
there is no consultation bureau for health/wellness. you are encouraged to take your child to his/her pediatrician once a year for a well-child exam. this visit will do all the things you mentioned, track growth on a growth chart, check eyes, ears, large and small movements, etc... You can ask for dietary advice at this visit. However, unless your child requires a vaccination or booster shot for entrance to school, these exams are not mandatory. Health insurance pays for the visit, but if you don't have insurance you would have to pay out of pocket. poor or low-income families may have access to care at a local free health clinic designed to serve the needy. since i haven't used a clinic, i don't know what they are like.
Pre-school is not required in the United States. I'm not sure if this issue is federal government or state government. I know that my state, Kentucky, is looking into mandatory pre-school provided in the public education system. It is not yet a reality. Our public school system offers pre-school for students who have a disability or delay. This can range from mild speech issues to autism to physical disabilities. Students with no disabilities are admitted if all the other spots are not filled, but the public pre-school program is very very small so there aren't a lot of openings. Most folks pay for pre-school if they wish their child to attend. Typical programs offered are often through local churches. My children went to pre-school for 3 hours 2 days a week at the age of 3/4. Then at the age of 4/5 they went 3 days a week for 3 hours. Pre-school activities are similar to what you mentioned above, although my children's school did provide a specific curriculum in which numbers and the alphabet were taught in addition to units throughout the year such as weather, dinosaurs, seasons, all about me, careers, etc....
Once in the school systems grades are kindergarten, then 1st through 12th grade. You graduate at the end of 12th grade and then may go on to college/university if you wish.
Hope this helps! I must get back to my own research.