What is the evidence for preferring charity on the eve of Friday and on Friday itself, and why does this contradict the impermissibility of singling out Friday for charity?

As for the disagreement between the questioner and his companion, there is no need for it, as this issue is one of juristic independent reasoning (ijtihad). There is no harm for someone who finds one opinion more preponderant or who follows a scholar who issues a fatwa supporting it. Fatwa number 101037 mentioned both opinions and favored permissibility, and it is in agreement with Fatwa number 74820. Fatwa number 60948 did not state that it is impermissible to specify Friday for charity; rather, it mentioned that nothing authentic was narrated from the Prophet, peace and blessings be upon him, regarding it. Its statement was made as a precaution and to avoid disagreement. The preponderant view on this issue is explained in Fatwa number 101037, and there is no contradiction between the fatwas.