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.