What do you call a word that has multiple senses or parts of speech in one sentence?

I think you're referring to the term zeugma.

zeugma noun A figure of speech in which a word applies to two others in different senses (e.g. John and his driving licence expired last week). - ODO

The dictionary entry also references the term syllepsis:

syllepsis noun A figure of speech in which a word is applied to two others of which it grammatically suits only one (e.g. neither they nor it is working). - ODO

Both terms are referenced in the following entry:

In rhetoric, zeugma (/ˈzuːɡmə/ ... or /ˈzjuːɡmə/; from the Ancient Greek ζεῦγμα, zeûgma, lit. "a yoking together") and syllepsis (/sɪˈlɛpsɪs/; from the Ancient Greek σύλληψις, sullēpsis, lit. "a taking together") are figures of speech in which one single phrase or word joins different parts of a sentence. - wikipedia

Here are a couple of examples from yourdictionary.com:

• "The farmers in the valley grew potatoes, peanuts, and bored." (Wunderland)
• "She opened her door and her heart to the orphan." (Wunderland)

The first example of the OP 'newspaper' is further defined in Wikipedia as a 'semantic syllepsis'. The article also lists four types of Zeugma; then a Diazeugma, a Hypozeugma, a Prozeugma and a Mesozeugma - thrilling stuff.

Zeugma (often also called syllepsis, or semantic syllepsis): a single word is used with two other parts of a sentence but must be understood differently in relation to each.

Example: "He took his hat and his leave."

My second example 'dust' might be cheating, I am not sure; because the second mention of dust is an ellipsis.

In my mind, the example sentence illustrates synecdoche...

synecdoche (n). A figure of speech in which a part is used to represent the whole (The American Heritage Dictionary of the English Language, 3rd ed., 1992)

...because the physical newspaper is being used to represent the entire organization that produces it.