Hun reduserer avfall hjemme ved å bruke gamle knapper på nye klær.

reduserer is the present tense form (3rd person singular) of the verb å redusere (to reduce). Norwegian adds an -r ending to verbs in the present tense:
• jeg reduserer
• du reduserer
• han/hun reduserer
redusere is the infinitive form (“to reduce”), so you need reduserer to say “she reduces.”

Norwegian omits the indefinite article for plural nouns and uncountable nouns when speaking generally:
• avfall (trash/waste) is uncountable.
• knapper (buttons) is plural indefinite (some buttons).
• klær (clothes) is plural indefinite (some clothes).
If you wanted to refer to specific items, you could use the definite forms (e.g., de gamle knappene), but here it’s general.

hjemme is an adverb meaning “at home.” In a main clause, Norwegian word order is typically Subject–Verb–Object–Adverb:
Hun (S) reduserer (V) avfall (O) hjemme (Adv).
So hjemme naturally comes after avfall. You could front it for emphasis: Hjemme reduserer hun avfall, but the meaning stays “She reduces waste at home.”

ved å + infinitive expresses the means or method: “by doing something.”
– ved å bruke = “by using.”
for å + infinitive expresses purpose: “in order to do something.”
– for å bruke = “in order to use.”
In the sentence, ved å bruke gamle knapper means “by using old buttons” (method), not “in order to use old buttons” (purpose).

Yes. i hjemmet (“in the home”) is a noun phrase, and hjemme is an adverb. Both mean “at home” here:
• Hun reduserer avfall hjemme
• Hun reduserer avfall i hjemmet
hjemme is shorter and more common in speech, but both are correct.