Esperanto compound verbs (esti + participle) work like English compound verbs (e.g. "am doing" or "have done") - the transitivity of the compound is like that of the main verb.

So "He touches me" has "me" in the objective case and so does "He is touching me".

Similarly, "Ŝi finas la laboron" has "la laboron" in the accusative and so does "Ŝi estas finanta la laboron" (or "Ŝi estis finanta la laboron").

I don't think so, as a much closer, more accurate translation is possible.

Your sentence is simple past tense, but the sentence to translate is continuous past tense. Translating the English continuous tense to the simple tense is OK for, say, German, where there is no continuous tense. But you shouldn't do it when there's an exact translation.

I think so. The continuous tenses are there in Esperanto but they aren't used all that much in daily life.