Learn French: J'aimerais que mon arrière-grand-mère trouve cette archive. - I would like my great-grandmother to find this archive.

No, because the verb trouver (to find) takes a direct object. The pronoun en is only used to replace inanimate objects when the verb is followed by the preposition de. To say 'find it', you use the direct object pronoun la: J'aimerais qu'elle la trouve. You would only use en with a verb like parler de (to talk about): J'aimerais qu'elle en parle (I would like her to talk about it).

Why does this sentence use que instead of just an infinitive, like 'J'aimerais mon arrière-grand-mère trouver'?

English uses the structure 'I would like [someone] to [do something]', but French does not. When expressing a wish or preference for someone else's actions, you must use aimer que or vouloir que followed by a new clause. This also triggers the subjunctive mood, which is why trouve is in the subjunctive form here (even though it looks exactly like the present tense for regular -er verbs).

Why is it mon arrière-grand-mère if a great-grandmother is feminine?

Even though the noun is feminine, the word arrière starts with a vowel. To avoid the awkward sound of two vowels crashing together (which would happen with ma arrière), French uses the masculine possessive adjective mon before any singular feminine word that begins with a vowel sound.

Since archive starts with a vowel, why do we use cette instead of cet?