Producer prices are more volatile than consumer prices. The CPI includes services components - which are more stable than goods - and the PPI does not. Wages are a bigger share of the costs at the retail level than at the producer level. Commodity prices react more quickly to supply and demand. Volatility is higher earlier in the production chain. Food and energy prices are major sources of volatility, hence, the greater focus on the "core PPI" which excludes these two components.
The bond market rallies when the PPI decreases or posts only small increases, but bond prices fall when the PPI posts larger-than-expected gains. The equity market rallies with the bond market because low inflation promises low interest rates and is good for profits.