You are essentially correct. The phrase おやすみなさい（お休みなさい・御休みなさい if you are choosing to write with the characters）is a conjugation of the verb 休む (やすむ：rest, take a day off, lie down etc.), from which the noun 休み（やすみ）is also derived. So, the link is a direct one.
To be specific, おやすみなさい is a (one of various!) polite imperative form of the verb 休む. The なさい and ...
The から is a 助詞(particle), meaning "after~" (or "from~" "since~").
(The plural suffix ら should be attached to a noun or pronoun.)
Is the reason the dictionary claims のもとで/もと is a noun
I don't think they claim のもとで is a noun. They're saying もと (下) is a noun, and that ～のもとで can be used to mean "under~~".
I think you could ...