I suggest reading the Wikipedia article on intelligence to get some of your questions answered.
I personally like this use of the term: “A very general mental capability that, among other things, involves the ability to reason, plan, solve problems, think abstractly, comprehend complex ideas, learn quickly and learn from experience. It is not merely book learning, a narrow academic skill, or test-taking smarts. Rather, it reflects a broader and deeper capability for comprehending our surroundings—“catching on”, “making sense” of things, or “figuring out” what to do.”
Among your observations is that schoolwork is commonly taken as a reasonable way to measure intelligence, and of course it does not do that well. Nevertheless there is a rough correlation between groups of high intelligence and performance in school or on standardized tests, much the same way that a cohort with superior athletic ability would be expected to perform well on the basketball team even though individual outliers exist.
There has been a general trend away from formally measuring intelligence. Why this is so would be a topic of debate all by itself. In my opinion there is a fundamental distaste for doing so because the effect is to stratify groups within a society determined to be as egalitarian as possible. To date all measurements of intelligence show a substantial disparity among populations, and since the upper limit for a given individual’s intelligence is determined by heredity, conversations about the performance of a genetically-related population rapidly deteriorate into arguments over nature versus nurture as the fundamental reasons some groups (in the US, groups self-described as black, for example) do so poorly on formal intelligence tests and standardized testing in general compared to other cohorts (self-described asians, for example).
You will see arguments that range from “there is no such thing as intelligence” to “there is no good way to measure intelligence at all” to “there are so many kinds of intelligence that current tests are totally inadequate.” There is a kernel of truth in each of these arguments, but generally speaking what you see around you everyday is a reflection of that ability we call “intelligence.” At school, highly intelligent people, on average, get better grades, have a higher capacity for storing and processing facts (even though by itself, that’s not “intelligence”) and score better on tests. In the workplace, highly intelligent people, on average, make more money. Cool stuff you use everyday was conceived of, and developed, by highly intelligent people. Super cool stuff like a space station or a collider have super intelligent folks pulling them off.
The commonest misuse of the term “intelligence” that I see is a confusion of “intelligence” with “knowledgeable” and the use of its opposite–the term “dumb”–with “ignorant.” An individual ignorant of a fact is not necessarily stupid, and an individual in possession of a fact is not necessarily smart, even though there is typically a pretty good correlation between high intelligence and the ability to learn, process and regurgitate a large volume of facts. In schoolwork in particular, where the teachers themselves are not necessarily required to be from the highly inteliigent cohort, it’s not unusual to find outliers–highly intelligent people whose genius goes unrecognized by teachers of ordinary intelligence.