I made poolish and refrigerated it for 24 hours brought it to room temperature and made a loaf which came out great. If this was my sourdough, I would then discard half, feed with water and flour and sit in warm area until it was ready.
If I use some poolish do I also discard half, feed with flour and water and refrigerate for 24 hours again?
From my understanding a poolish is made up of flour and water at 100% hydration, with a sprinkling of yeast. The purpose of the poolish is to develop flavor compounds that will be added to the final mix. The yeast consume elements of the water/flour mixture and leave flavor and color in their wake.
In the case of a poolish you are not trying to grow yeast, in comparison to a levain where you are trying to grow bacteria. If you wanted more yeast you would just add more yeast, either fresh or dried.
I suppose if we could purchase lactobacillus sanfranciscensis and simply add it to a flour/water mixture we all wouldn't be fussing around with sourdough starters but that isn't the case. We have to grow and maintain our own cultures.
With SD, you need to keep / maintain a starter because it takes at least a few days for a new one to be ready to use. Poolish is ready to use after a few hours to overnight, so I see no real reason not to make a new batch before every bake.
And re. discarding starter, you might want to consider finding a method that doesn't require discarding. There are different possibilities depending on how and how often you bake.