...what are more or less the grammatical and vocabulary (+kanji) contents covered in those two subjects?
You will cover Chapters 1-6 in Semester 1, and the remaining chapters of Genki I in Semester 2. Whatever kanji is listed at the back of that book will be the only kanji you need to learn. Any other kanji are supplementary and unknown words found in passages will be given to you in tests.
This is not always the case, though. In contrast with Jun, Yasuhisa Watanabe (JP3/4) emphasises that you should be able to scan through a passage, ignore irrelevant information that includes kanji you have not studied, and pick out portions that are relevenant to the exam. I can't really remember if this is what we had to do in JP1/2.
(Also note that the exam literally takes the passages at the back of Genki and asks you questions about them. Helpful tip if you wanna get 80+. I'm not even joking, it's the same shit you do in class.)
... and I'm wondering if I can just take the entry exam and enroll in Japanese 3 directly. I don't know though if I know or experienced enough to join it.
If you can complete Genki I's workbook without any trouble, you can enter Japanese 3. However, being textbook smart does not mean you are ready for those classes. Some of the teachers in your seminars will only speak Japanese with you, so if your listening and speaking skills are not up to par, you will feel incredibly behind. This was the case with many of us who only took Japanese 1 and 2 at university. There is next to no effort by the teaching staff to help you learn how to speak Japanese on the fly. I had to resort to a tutor I could speak with so I wouldn't fail a one on one conversation assessment with the teacher.
If you are coming out of Japanese in high school you'll be 500x more ready for this aspect of the course than I was. Of course, this only applies to Japanese 3 and beyond. Japanese 1 and 2 does not require you to be able to listen and speak so quickly.
I might also ask what N level is the first two subjects (Japanese 1 and 2) are equivalent with?
We were recommended to take the N5 after Japanese 2. Completing the two Genki textbooks (that means taking Japanese 1, 2, 3 and 4) meant N4. At this point you could dabble in N3 if you've been studying Kanji with a SRS deck or WaniKani or whatever tool you use. N2 is only doable with a lot of independent study (possibly after the Minna no Nihongo series which is covered in Japanese 5+). No one ever needs N1 for a job.
ありがとございます
ありがと
うございます. Spelling mistakes like that is how they get ya in the weekly grammar tests.

Good luck!