Can I still get into an Ivy League school with no honors/AP? The schools in my country don't offer that.

And also you can't choose what classes you take. Everybody takes the same 23 classes.

No college or university requires AP courses, but you'll be competing against other applicants who have taken them if you apply to the most selective schools. If you don't take AP courses, you'll need other achievements to be as appealing to a college.