I made the mistake of translating this as "I saw her in the evening" - what would be the proper French version of that, maybe "Je l'ai vue la soirée"?
Yes, because the participle needs to agree with the direct object if it appears before the auxiliary verb.
Both "le" and "la" contract with "ai" to make "l'ai" so your only hint to the gender of the direct object is "vu[e]". If it is masculine (vu) then you can infer it is "le" - i.e. him - and if it is feminine (vue) you know it is "la" - i.e. her.
Yes, and "soiree" is used in English (usually without the accent), so "I saw her at the soiree" should be accepted - reported it.
In the 'hover above hints', a suggested translation was 'I saw to it...'. Did anyone try that? Can the phrase 'Je l'ai vu(e)...' really mean that, does anyone know?