Some key points to consider when analysing your sentences:
1. Using the terminology of the Dictionary of Japanese Grammar series, your sentences are of the format S1 toki S2 (Sentence 1 とき Sentence 2).
2. According to that dictionary's explanation (p493 of 'A Dictionary of Basic Japanese Grammar'), when S2 is past tense and S1 expresses a state, you can ...
I am not a Japanese grammar expert but just a native Japanese speaker.
I think you can say 「ローマに住んでいる時、子供でした。」.
I personally think there is no difference between the meanings.
However, as the comment of the original question mentioned, the sentence 「ローマに住んでいる時、子供でした。」is a little bit wired even though it is ...
見つかる is intransitive, "to be found/located". 見つける is transitive, "to find/locate".
A: 見つかりません = "it has not been found (so far / yet)."
B: 見つかりませんでした = "it was not found."
C: 見つけません = "I don't/won't find it."
D: 見つけませんでした (= "I didn't find it.")
見つかる is the normal verb used when you talk about the result of a search in Japanese. We don't usually say C and ...