And in terms of the volatility. I think the best way to learn them is kinda parrot fashion. I drew it out as two rows of ups and downs. For example. Volatility increases. Call option(up), callable bond (down), OAS (down), Overvalued. Volatility down, call option (down), callable bond (up),OAS(up), undervalued.
For puts: Volatility(up), option(up),puttable bond(up), OAS(up), undervalued. Volatility(down), option(down), puttable bond(down), OAS(down), Overvalued.

There are some general relationships here like the OAS will move in the same direction as the puttable or callable bond. So if you can figure out how the volatility effects the option and the bond then you know what the OAS will be. General rule is that you can consider the market price of the bond as being fixed. So if the price of the putable or callable bond increases you will need a bigger discount rate to bring it back down to market value, thus a bigger OAS.