Do Japanese Student councils have actual power (punishing students, revoking clubs, etc)?

Question

Several anime have an Absurdly Powerful Student Council (warning, TV Tropes link, don't get sucked in). In the most extreme versions, the student council is trying to take over the world.

Even in less extreme examples, the student council...

1. Punishes students for breaking rules or dress-code, sometimes outside of school grounds

2. Closes a school club to hurt the main character, or doesn't allow a club to open even with sufficient student interest

3. Has exclusive access to special libraries and other assets that would benefit all students

4. Has lavish "student council only" rooms that are off limits to all other students

5. Decides which clubs get the most funding (and uses it to make people do favors for them)

6. Has the keys for every door in the school, including dangerous places like the trash compactor, the furnace, and the roof.

7. Invites other students to join the council without a vote

Do Japanese student councils do any of this, or have other actual powers an American student council could only dream of?

EDIT

In American TV shows about high-schools or universities the villain is usually an adult teacher like Sue Sylvester from Glee or Ben Chang from Community, or a bully like Biff from Back to the Future. I do not believe I've seen an American TV show or movie where anyone on the student council was the primary antagonist.

Answer 1

TL;DR: No, real-life student councils do not have absolute powers and aren't as powerful as depicted in anime or manga.

Long answer

Japanese school life tends to be much more hierarchical and organized, with students being put in charge of far more of how things operate and thus the concept of student council comes. So first, let's define student council in Japanese schools. According to Wikipedia:

[...] student councils were added to Japanese schools after World War II. In Japanese schools, students in a class stay together as a cohesive set in the same homeroom for most of the day. Each class has one or more elected representatives who reports to student council. The student council consists of members who are elected by the student body. The council is often responsible for organizing events such as the culture festival, sports day, and class field trips. The council also oversees the school clubs, and has absolute influence on school and club policies.

Note that it says it can influence school policies and rules but that doesn't mean that they have the absolute power to set and govern school rules. Those are done by schools administrations. To be more precise¹:

In anime no one is cooler or more highly revered in high school than the student council. They are are held at a higher prestige, respect, and hold actual authority over other students. While in reality this is just not true at all. The student council does not have authority to make decisions when it comes to school rules or how the school operates. The school administration is held at higher regards than the student council, and age is just as important in regards to authority. One of the pole participants who was part of the student council in high school chimed in on the situation, “I was a member of the student council, and we didn’t have much actual authority.”

So, despite student council being depicted as the absolute power-wielding entities who can revolutionize the world, in real-life they have, more or less, very little power. But don't get me wrong! They are still very strict and disciplined, and at least hold powers to control club activities/policies and set all sorts of classroom rules. In some schools, they are given the power to set and run events like field day or cultural festival and organize class trips. Please read ref. 3 and ref. 4 for an exhaustive discussion.

References

1. Japanese Students Reveal Differences Between Anime High school And Real Life
2. 4 ways anime high schools differ to real-life Japanese high schools
3. What Are Japanese Student Councils REALLY Like? (Here, it points out how Japanese student council works differently from American schools. It also notes that Only Yesterday and His and her Circumstances have depicted student council accurately)
4. Japanese Student Council: Differences between Anime and Real Life (You can find an exhaustive explanation, discussing every differences by pointing out every depiction shown in anime and differentiating them from real life portrayal).

Answer 2

Kind of a weird question at least as of the current revision because I had the exact same question earlier this year and checked that tvtropes article for the answer and it's exactly there

Note that in Japanese schools, this is an exaggeration of Truth in Television, as the Student Councils actually do have power over the approval and funding of student clubs. School Clubs Are Serious Business is a real thing in Japan, as they're meant to prepare students for the high-pressure work environment into which they'll eventually be dumped, and students are strongly encouraged to join at least one club. And just like in their future jobs, students are expected to show utmost loyalty to their club: once you join, you're there until you graduate. Likewise, in private schools in the United Kingdom, prefect councils often perform the same functions that an honour council fulfills at a university, and any year's prefects are assigned to keep order in the teacher's absence. Are you starting to see the clout Student Council can hold? (More details in the Real Life section below.)