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Sorry but "A" can not be a change in producer surplus, if it was already part of it from the start. The change in PS is the space the area beneath the price Pw, and between S' and S. This is fairly clear!
At price p1 (which is the pre-subsidy price), Q1 is the quantity produced---so revenue of P1xQ1. A is not part of producer surplus at that price,quanity combination. After the per-unit subsidy, quantity Q3 is produced with total revenue from market sales of Q3xP1. Firms receive A+B in subsidy payments. B is an additional marginal cost above and beyond the market revenue. So A is a indeed the increase in producer surplus inclusive of the subsidy payment.
No. "A" is part of the subsidy payment (presuming that all production receives the subsidy). The original production subsidy is not labeled in this graph. It was the triangle between the world price and the domestic supply curve.
