The major Sunni hadith collections were compiled before the major Shi'i collections primarily because the Shi'is had the 12 Imams to rely upon to answer their questions, and the 12 Imams are considered to be absolutely accurate in their transmission of hadith from the Prophet (S).
Additionally, Shi'is were subject to intense persecution for much of their early history, and so it was more difficult to compile such books.
As for the correctness or incorrectness of hadith, this is a subject discussed in depth in the study of hadith, and I would recommend that you read some books on the subject. Here is a suggestion: https://www.amazon.com/Introduction-Hadith-Abd-Al-Hadi-Al-Fadli/dp/1904063470
However, one area that the Shi'is had an advantage is that hadith were being recorded during the lifetimes of the Imams (in that the major compilations, such as al-Kafi, were not the first records of hadith), whereas Sunni recording of hadith, for the most part, only started a hundred years after the Prophet (S). (I use the term "Sunni" loosely here to refer to what eventually became Sunnism)