Questions & Answers about Я попросил, чтобы брат не трогал стекло.
Why is трогал (touched) in the past tense here when the English translation says "not to touch"?
The conjunction чтобы forces the verb in its clause to be in the past tense, even if the request is about the present or future. Whenever you report a command or request using чтобы, you must use this past tense form.
The current lesson is about using the perfective aspect for negative warnings. Why is трогал in the imperfective aspect here?
This is a reported request not to do something deliberately, not a warning about an accident. In Russian, direct or reported negative commands ("don't do that!") almost always use the imperfective aspect. The perfective is saved for warnings about accidental mishaps, like in Смотри, не разбей стекло (Watch out, don't break the glass).
In English we say "I asked my brother". Why is брат in the nominative case here rather than being the object of "asked"?
Because чтобы starts a completely new clause with its own subject and verb. Literally, this sentence says "I asked that the brother not touch the glass." Since брат is the subject of the second half of the sentence, it takes the nominative case. If you wanted him as the object, you could say Я попросил брата, чтобы он... (I asked my brother so that he...).
Is стекло in the nominative or accusative case here?
It is in the accusative case, because it acts as the direct object of the verb не трогал. For neuter nouns ending in -о like стекло, the accusative form looks exactly the same as the nominative form.