The issue of verb construction between languages is always a messy one. However, in this case it’s easy to explain: “descendre de” does not quite mean the same thing as “descendre” alone.

It implies something like “she comes from the top of the hill” (it’s the TLF’s sense 1.A.1.). You CAN say “elle descend la colline en courant”, but that covers a broader sens of meaning, i.e. “she comes down the aside of the hill” (I.e. TLF’s sense II.B.).

Well, “to climb up” is “monter” or “monter sur”, while “to climb down” is “descendre” or “descendre de”.

There is a difference in meaning. In the forms with a preposition, you aren’t on the object before (or after) the action. While in the forms without the preposition there is the idea that you stay on the object but are moving up or down.

So you’ll “monter sur une chaise” or “descendre d’une chaise” (you can’t stay on the chair while moving up or down). For the hill or the stairways, you can say both, depending on what you are insisting (the movement — the form without a preposition — or the result — the form with a preposition). So you’ll rarely use “monter sur les escaliers” or “descendre des escaliers” as, excepted in describing children games or when you want to reach something, what is usually important is the movement, but for the hill both forms are commonly use and will suggest different mental images.

De peut avoir le sens de spécifier une origine de quelque chose ou quelqu’un. Votre premier exemple manque simplement une origine spécifique. Par exemple « Elle descend du grenier par les escaliers ».

De can also have the sense of from in the sense of originating somewhere. Your first example simply lacks a specific origin. It could be reformulated as “She came from the attic using the stairs”.

Example: “I come from England”
«J e viens d’Angleterre »

Example: The gift-tag on a present “From: George To: Anne”
L’étiquette sur un cadeau dit « De: George Pour: Anne»

Your examples:
« Elle descend les escaliers »
“She descends the stairs”

« La femme descend de la colline en courant »
“The woman runs down from the mountain”.