When a card is moved out of community and lands in a row where a mousetrap kills it, do the symbols still trigger? (Draw faction and draw community) Or is it dead before those trigger?

Yes, any time you draw a starting card, all actions trigger but the card would be killed.

Even ones moved by the starting card?

Draws would be the only actions that could happen when a card is moved by the starting card and those occur when they break the neutral area line. Once they land in the row, they are killed but the draw and/or faction action would still occur.
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