She'll take the container down to Bluff House with her. We often use "down to" or "up to" to express a change in location even if there is no physical difference in elevation or a north/south distinction. Similarly, people talk about going "up the street" or "down the street". If there's an obvious hill, they might use the appropriate directional term, likewise if there's a clear north/south distinction. We're going down to Florida for a vacation; we're going up to Canada for the weekend.

In this context, "down with her" is a slang phrase that means "take with her". This usually is to a specific location, e.g. "I'll take a deck of cards down with me to Maria's house for game night."

"She’d take the container [she’d culled out for Eli] down with her." The phrase "she'd culled out for Eli" is acting as an adjective here, modifying "container." Which container? The container she'd culled for Eli. Let's try a different phrase here, and then let's just take the phrase out completely. "She'd take the container of stew down with her." And lets remove a little more and then build it back up. "She'd take the container down." She'd "take." Take what? the container. Where will she take it? down. "She'd take the container down with her." How will she take it? With her. She's going downstairs anyway, it seems, she might as well carry the container with her. And finally: "She’d take the container she’d culled out for Eli down with her." Which container? The one she'd culled out for Eli. I don't know the proper grammatical name for what's happening here, but the phrase is working together with the word "the," too. "The" means "I'm thinking of one particular container," and the phrase tells us which particular container it is.