Can you play the Sicilian Defense after white not playing 1.e4, or is it even called that in this case? Also, after starting with the English Opening, can you only play reversed sicilian if black's first move was 1.e5?
You cannot force the Sicilian Defense after a White move other than 1.e4. The Sicilian Defense is defined by 1.e4 c5, not by 1...c5. Chess openings in general are named based on the moves played by both sides, not by just one side.
It is possible sometimes to end up in the Sicilian defense by different routes, e.g. 1.Nf3 c5 2.e4, which arrives at the same Sicilian position as 1.e4 c5 2.Nf3. But White needs to cooperate for this to happen.
Similarly, as you guessed, a reversed Sicilian implies that White and Black have pawns on c4 and e5 respectively. If Black doesn't play ...e5, it's not a reversed Sicilian. But again, he may play ...e5 later than move 1, transposing into a reversed Sicilian line at that time.