Does the Prophet's placing of his hand under his cheek mean the palm or the forearm, and what is the wisdom behind that if placing the palm leads to the arm falling asleep?

The term "hand" in the hadith refers to the palm. This is indicated by the narration from Al-Baghawi and An-Nasa'i: "When he went to his resting place, he would place his right palm." This hadith was graded as hasan (good) by Al-Baghawi and sahih (authentic) by Al-Albani.