But I don't know what to do here, there are wires going into a block between the blue plate and the heatsink, and it is covered in glue. I don't know if it's alright to remove the heatsink, or how to power it by batteries instead of USB. Any suggestions?
It is a peltier cooler (thermoelectric cooler, TEC). You just run current through it and one side gets cold and the other gets hot. You cant really expect to power it with batteries as it uses a lot of current, you could probably run several amps through it if you wanted, but it just depends on your heat load.
TECs are best in line powered lasers as you always have a good power supply, but you can run them from batteries if you want - you could use an LM317 to regulate power to it
So if it could power with batteries they probably wouldn't last long eh.
I popped off the blue cover and cannot remove the heatsink. But I did split the peltier in half and can see hundreds of little squares and connectors, some are broken in half, missing, attached to the opposite side, it doesn't seem to cool anymore when I plug it in
yeah, it sounds like you ripped the actual TEC in half instead of just removing the top "cold plate" thing. There should be a bunch of little things that have a piece of ceramic or something like that which they are sandwiched between. No big deal if you want another one to play with they cost like 10 dollars of some websites.