BBava Batra 124b, for instance, has Rav Nahman quoting "the remaining books of Debei Rav." How many were there? What were they exactly? If they contained written halachot, why were they allowed? (Assuming that it doesn't just mean the Sifra/Sifrei as we have them.) Articles references/modern sefarim would also be welcome.
An amora is quoted (Vayikra Rabbah 7:3) that elementary students first began their studies with the book of Leviticus, “the young are pure and the sacrifices are pure, let pure be occupied with pure”.
In the introduction to his magnum opus Louis Finkelstein observes (Vol. 1 p. 4) that when students graduated to studying biblical exegesis they would commence with Torat Kohanim and therefore TK was nicknamed ‘Sifra dve rav’ (the students’ book) whereas the other Pentateuchal midrashim were referred to more generally as ‘shar sifre dve rav (the other/rest of students’ books).
Sifrey is a plural. You can see in Rishonim that they quote The Mechilta DeRashbi also as "Sifrey". סתם ספרי רבי שמעון (Sanhedrin 86a) Rav wrote Sifra and Sifrey. Sifre is generally according to RaShBY opinion. see here in wikipedia. The main book (the largest) is on Vaykra, it is Called "The" book, Sifra. Sifrey, the remaining books of Rav on Bamidbar, Devarim and may be Shemot following the remark above concerning mechilta de Rashbi.