Learn German: Der fleißige Nager verschwendet keine Zeit. - The hardworking rodent wastes no time.

Yes! This is a great example of a transitive and intransitive verb pair, which is the focus of this lesson. verschwenden (to waste) is the transitive, weak (regular) verb. It takes a direct object—here, the time being wasted (keine Zeit). verschwinden (to disappear) is its intransitive, strong (irregular) partner. It doesn't take an object. In German, it is common for a weak verb to describe causing the action of its strong verb partner: "wasting" something causes it to "disappear."

Is Nager the standard word for "rodent"?

Yes, der Nager is a common and slightly casual word for rodent. It comes directly from the verb nagen (to gnaw or nibble). You will also frequently see the synonymous compound noun das Nagetier (literally "gnawing animal").

Since Nager is a masculine noun in the nominative case, why does the adjective end in -e (fleißige) instead of -er?

Because the definite article der is already doing the heavy lifting by showing the masculine nominative case. When the article clearly displays this grammatical information, the adjective takes the "weak" declension ending, which is just -e. You would only use the strong -er ending if that definite article were missing, such as in ein fleißiger Nager.

In English we might say "does not waste time." Why is it verschwendet keine Zeit rather than verschwendet Zeit nicht?