If my opponent has no cards in hand and I cast Nezumi Shortfang, does Nezumi automatically flip or do I have to pay to make Nezumi flip?
The flip is not a separate trigger, it is part of the effect of the activated ability. So you do have to pay and activate it. You don't need them to have a card in hand to do so; if they don't, the discard part won't do anything, but the Shortfang will flip.
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In general, to flip Nezumi Shortfang, you must pay to activate Nezumi Shortfang's ability (and the ability must resolve with a legal target [C.R. 608.2b] and the opponent must have no cards in hand after they discard a card if any). Note, however, that the instruction to "flip Nezumi Shortfang" can be carried out even if the opponent had no cards to discard due to that ability (the ability doesn't say, for example, "If that player does and they now have no cards in hand...") (C.R. 609.3).
Flip card question