What is the reason for the variation in the subject pronouns from singular to plural in the story of Musa and Al-Khidr in Surat Al-Kahf, and specifically, why did Al-Khidr use the plural form "فخشينا" (so we feared) and "فأردنا" (so we intended)?

Al-Khidr (peace be upon him), in the story of Musa, explained the reason behind his three actions. He attributed the scuttling of the ship and the killing of the boy to himself, then attributed the building of the wall to Allah. This was out of good manners with Allah, so that what appears to be evil would not be attributed to Him, even though everything is by His decree.

Regarding his reporting about himself in the scuttling of the ship using the singular first-person form ("so I intended") and then using the plural of majesty in the killing of the boy ("so We intended"), Al-Shawkani believes that the plural of majesty was used sometimes to speak of Allah's blessing upon Al-Khidr, and sometimes out of humility.

It could be that the singular was used in the scuttling of the ship because it was a minor matter, whereas the killing, due to its gravity, necessitated the plural form.

As for attributing the building of the wall to Allah, it is because the two boys reaching maturity is something beyond human capability; it is an act of Allah alone.