when are they checked for double spends or false coins? Is that when the miners put them in the block?
The miner put transactions in the block that are valid for the current blockchain. When an other pool finds a block that makes a transaction invalid, it will be exluded from the next atempt to find a block. Also if 2 or more transaction occour, which invalidates each other (double spend), each miner chooses one to include in his block