The most important thing to understand about the American educational system is that there is no American educational system.
Public schooling is the responsibility of the states, not the national government. (“Public” meaning government-run, not like the meaning of “public school” in the UK).
Further, most states delegate the day-to-day business of running schools to local school districts, typically run by a board which may have elected or appointed members. So, standards about ages, subject matter, performance, teaching requirements, and the level of autonomy of school boards vary considerably among the states.
The federal government does provide significant subsidies to state and local governments to run schools, and these subsidies come with federal requirements attached. This is the only way that Congress can affect local schools directly, barring some other area of federal jurisdiction that also happens to affect schools.
Typically, students will start kindergarten or first grade at around age five or six. Many parents will send their children to nursery school before then, and there are a tiny number of public nursery schools, though most are run by private organizations. Not all school systems have a kindergarten; some start at first grade.
The organization of school levels varies enormously. My public school district had four elementary schools teaching K-6 (that’s ages 5-12, roughly), a single middle school serving grades 7 and 8, and a high school serving grades 9-12. Other places but grades 6, 7 and 8 at the middle level, others 5, 6, 7, and 8. Still others put 6, 7, 8, and 9 at the middle level and 10-12 at the high school.
Most people start first grade at six years old, so add five to the number. Most people graduate high school (12th grade) at age 18.
Generally, a 10th-grader. In a four year high school, grades are referred to as freshman, sophomore, junior, and senior. This pattern is repeated at four-year colleges.
Depends what you mean by “serious.” 
Beginning around middle school in most places, students will have to take final exams (either twice yearly or once at the end of the year) to prove proficiency in each subject. Many states have statewide standardized exams for high-school level subjects; in New York, these are called the Regents Exams. You can’t graduate without a certain number of Regents credits. Many states use standardized tests for benchmark basic skills in elementary school, as well.
For college admissions, the most important tests are the SAT and ACT, both of which are optional (most college-bound students take only one.) Both of these tests are given by private organizations funded by colleges.
Additionally, most high schools offer some number of Advanced Placement (AP) courses – these are college-level courses for smarter students and culminate with an AP exam at the end. If you score high enough on the AP exam, most colleges will accept that as class credit. Finally, the SAT people also offer optional subject-specific tests (at the non-AP level) which used to be called SAT-II’s. Colleges like to see a few good scores on the SAT-II.
There are minimum requirements to graduate high school in every state. In New York, when I went to high school, those requirements included four years of English and history classes, at least two years of foreign language classes, a certain level of math proficiency, and various science topics including biology, chemistry, and earth science. All of these required the above-mentioned Regents exams.
In America, when your teacher raps you on the knuckles for making noise, you can sue them for ten billion dollars.
