Learn Finnish: Minä torjuin hänet ja käytin huonoa tekosyytä.

It comes down to the verbs and the nature of the action. Torjua (to reject) here describes a complete, finished action, so the personal pronoun takes the accusative form hänet. However, the verb käyttää (to use) is a partitive verb, meaning its object almost always has to be in the partitive case, hence huonoa tekosyytä.

If the current lesson is about ei kumpikaan, how would I use it here to say "I rejected neither of them"?

You would say En torjunut kumpaakaan. Remember that in negative sentences, the object must always be in the partitive case. So, instead of the basic form kumpikaan, you must use its partitive form kumpaakaan.

What words make up the compound word tekosyytä?

It is the partitive form of tekosyy (excuse). This is made from teko (an act or deed) and syy (reason or cause). Put together, a "deed-reason" translates to a made-up reason, or an excuse.

Why does käytin only have one t, while the dictionary form käyttää has two?

This is standard consonant gradation. When you add the past tense marker -i- and the personal ending -n, you close the final syllable (käy-tin). A closed syllable triggers the weak grade, forcing the strong double tt to drop to a single t.