Re: Breastmilk storage after starting to drink it
The La Leche Legue clearly states on their breast milk storage page that it is not known whether or not thawed milk that is left over from a feeding can be kept refridgerated until the next feeding. Do so, I imagine, would be at your own risk (I do it, but I make sure it's in the fridge).
Room temperature fresh milk can keep for 10 hours (not left over after a feeding), in the fridge for 8 days, in a freezer 3 months, and in a deep freeze for 6 months.
As for keeping it in a diaper bag - invest in some small ice packs. I keep two in freezer, and these, in my insulated diaper bag bottle compartment, are enough to keep BM cold (as if it were in the fridge) for several hours in warm Florida weather.