Added cories to new tank, fish appeared to go into shock, improved after removal...

Water from an established tank wont do anything as all the bacteria that are beneficial are in the filter media. U need to start from scratch..get the tank out of the bathroom and cycle using media from an established tank. All those cories will die if that tank isnt cycled u wont just be "sacrificing" a few.

Not true.. In a cycled tank, everything is a bacteria host.. Gravel, decorations, glass, whatever..

Go to your LFS (not Wal-Mart!!!!!!! Never buy living animals from the same place you buy toilet paper..) and buy a few lbs of gravel from their tanks. Bring it home, break that tank down and replace the gravel. This will get you 90% of the way to cycled. The bacteria in the gravel will handle anything those cories can produce, and the rest of the tank will catch up within a couple weeks. If you can get filter media, that'll help too, but gravel is your best bet for a quick fix.
 
Corydoras are hardy fish so they should be alright in the tank.But you might lose some.

That is some of the worst advice I've ever seen, cory's are very sensitive to ammonia, which spikes in newly setup tanks when cycled properly, they are a very bad choice for a fishy cycle.

Go to your LFS (not Wal-Mart!!!!!!! Never buy living animals from the same place you buy toilet paper..) and buy a few lbs of gravel from their tanks. Bring it home, break that tank down and replace the gravel. This will get you 90% of the way to cycled.

LOL, I totally agree about the TP, but don't forget to scrub that tank out with vinegar and rinse it well before you refill it and start again!
 
Sorry i miswrote, what i meant to say is that established filter media would be far more beneficial than just tankwater. Usually I add some established media and some gravel encased in nylon from another tank to get a quick cycle.
 
In addition to the above, you need to slowly acclimate new fish to your water by adding small amounts to the bag every 10 minutes or so. Do that until you have ~2/3 new tank water to lfs, some have very different water ie nitrates & TDS. Acclimation is more than temp.
 
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