Anime's depiction of student council, going on the roof top and K-On club - If you didn't notice already from the title, Fiction.
Lets start out with student council. So in anime it seems like most student councils are portrayed as an all powerful group of students that can do as they please, make rules and disciple other students. In actuality this is fair from the truth. Now while Japanese student councils have a more meaningful role then American student councils (not to be harsh to anyone who was in one, but are more or less... pointless) they simply aren't on the level you see in anime.
They can not punish fellow students or decide on any disciplinary measures for something that is done. While they can set up some very simple, basic guidelines for students, they can not out and out make strict rules that have to be followed by the other students. The closest they can come to actually making rules is to come up with a rule or a rule change and propose it to the teachers, but the teachers have to agree to the change. In most cases as well they will often have to also be agreed upon by the student body. What they mostly seem to do is help with different events that happen during the school year. They will do an introduction ceremony for the new students, run in a sense an opening ceremony for the culture festival, host different assemblies like club introductions for the new students and things of this nature. Perhaps their most "official" moment of the year is the student councils yearly assembly where they will go over guidelines and talk about the events calendar and mention any big changes coming up. This is followed by other students asking questions, mainly why they can't do certain things, and the members of the student council trying their best to answer though most of the questions are out of their control and the teachers are the ones who would be best suited to answer them.
You do have your typical positions like President, Vice President, Secretary and the like and then also many leaders of different committees who will help out and have input on whatever committee they are in. Also they do meet often. Not everyday, but multiple days a week they will have meetings.
So when it comes to the student council it does do more then what you'd see in America, but no where near as powerful as you see in anime.
Now onto the next random bit and that would be going on the rooftop. I honestly think it would be difficult to find an anime that is set in school and doesn't have at least one scene take place on the roof. My question is how is this possible. I have never been to a school that doesn't have the roof locked at all times. Only twice have I ever seen a student on the roof. Once was for a year book picture where the homeroom got special permission to take a class picture on it and the other being when the bad students broke the window and hid up their during the sports festival. I'm guessing rooftop scenes make for good drama, but I've never once even heard of a open roof where students can go, eat lunch, relax or whatever. So this is pretty much just straight up fiction.
The last random fiction of the day is basically based on one anime in particular, but I guess also will get its influence from others and that being K-On is a small, niche, unpopular club. So pretty much in the K-On! anime the club is struggling to survive and in pretty much every other anime I've seen the K-On club never even seems to get background filler. Beck mentioned a light guitar club, but it seems like K-On is almost unheard of except for one anime that makes it out to be unpopular. Well if it isn't the most popular club you'll find now in high schools it's easily among the top of the list as well as one of the most rapidly growing along with the dance club. That's another thing, how have we not gotten a dance club anime? Oh well. When it comes to the K-On club it's just huge now. Students are lucky to get 15 minutes of practice time a day in the K-On room, cultural festivals will have an entire days lineup easily filled for concerts in the gym. It's just huge. So perhaps not the biggest "is that true" thing about anime, but a minor random one that seems much more fiction than fact.