All perfect praise be to Allah, The Lord of the Worlds. I testify that there is none worthy of worship except Allah, and that Muhammad is His slave and Messenger.
Ibn Khuzaymah, Ibn Hibbaan and Al-Haakim are among the leading scholars of Hadeeth, who memorized the Hadeeth and transmitted it to us. Therefore, saying that they are not reliable is a clear misconduct against them. No person who is engaged in learning knowledge would say such a thing. Consequently, you should avoid this dangerous attitude in dealing with the leading scholars of religion, like the scholars of Hadeeth, the jurists and others.
What these scholars were criticized with does not affect their trustworthiness at all. Indeed, no scholar –as far as we know –has ever said that the grading (classification) of these Imams on the Ahadeeth are rejected and should not be considered. Rather, their classification of the narrations has its weight and consideration, despite the criticisms of some who think that they were lenient in authenticating some narrations in some cases; these criticisms should not be taken for granted.
As regards Al-Albaani he is one of the most famous contemporary scholars [who is now dead] who was engaged in classifying the Ahadeeth and judging its chain of narrators. He was also criticized for his leniency in classifying some of the narrations.
There are many Imams of Hadeeth, who are reliable in their classification of the Ahadeeth, such as those whom you mentioned in your question, and those who came before them and who were of a higher rank than them, such as Imam Ahmad, Ibn Ma’een, Ibn Al-Madeeni, Abu Haatem, Abu Zar’ah, Al-Bukhaari, Ad-Daaraqutni and others. However, knowing their classification of the Ahadeeth may not be accessible to those who are not specialized in this field.
On the other hand, the classifications of Ibn Khuzaymah, Ibn Hibbaan and Al-Haakim, of the Hadeeth is easier because they have books that are categorized about the Saheeh (authentic) Hadeeth.
Knowing the classifications of the Hadeeth by Al-Albaani is even much easier, especially through the search engines of Ahadeeth on the Internet. He is one of the contemporary scholars who has classified the Ahadeeth the most, so there is no harm if you act according to his classification, whether he classified them as authentic or inauthentic.
Finally, you should know that classifying a narration as authentic is generally a matter of Ijtihaad (making the best judgment on a matter) which could be subject to a difference of opinion among the scholars of Hadeeth.
For more benefit, please refer to Fataawa 86276, 137786, and 299569.
Allah knows best.