If the pubkey is revealed,may someone use BSGS algorithm to crack the privkey and double spend the coins?

YES, there is a bunch of tools to do that some people running crazy crazy machines
the pubkey is not revealed yet and because of the small range 2^66..
you can try this buy yourself pick a random prvtkey 0x2AAA5555000FFFFFF run kangaroo and see!
The public key is already known information, but that is not enough to crack the private key, and if someone get's access to the private key first they need to spend the coins before be able to double spend it. But looks like you are lost in the topic, what op means is.
Let's say someone finds the private key from puzzles 66 and 67, if they spend the coins and make public the private key, som users could use that private key to make a double spend from the same coins and if they use bigger fees then they could steal those coins. But the way to avoid that is to not make public the private key until it has more than 1 confirmation.