Sig1- Depending on what animals you're trying to avoid in the woods the 9mm might not be enough. If you're carrying it in the woods for protection I would consider it the same as carrying it on the street and I wouldn't trust the magazine with the conversion barrel adequate. I would have a dedicated .40, .45, or a 10mm.
Sparks1957- yep it's all you need. If you are going from .357 to 9mm and using the smaller 9mm magazines the magazine will be a little bit loose in the frame and be more prone for bad feeds. The recoil spring will be just fine.
Like I said in my other thread all you have to change to go from .40/.357 to 9mm is a conversion barrel and it would be just fine for the range. If I was carrying concealed (which I do with my P229 .40) though I wouldn't trust the conversion barrel and the wrong magazine- I would carry a dedicated gun be it .40, .357, or 9mm but that's just me.