What is meant by the phrases "on the condition of the Shaykh," "the Two Shaykhs," and "they did not narrate it" in hadiths, and do they affect the authenticity of the hadith?

Yes, it diminishes the authenticity of the hadith. This occurs in one of two ways: either by reducing its degree of authenticity while it still remains authentic, especially when compared to a hadith agreed upon or one narrated solely by one of the two Shaykhs; or by reducing its authenticity to the extent that it becomes flawed (ma'lul) or anomalous (shadhdh), thereby moving from the category of authentic to weak. The fact that a hadith meets the criteria of the two Shaykhs does not necessarily mean it is authentic, as there might be a hidden flaw ('illah), an anomaly (shudhudh), or a disconnection (inqita') in its chain of narration (sanad).