Given the first half of a SHA-1 hash, can you predict the second half? I know this applies to secure ciphers, but does this also apply to hashes?
Not without knowing anything about the message.

Each 32-bit word of the hash is a function of the current state plus one 32-bit word of the expanded message.

If the message is longer than 448 bits (512-bit block less 64 bits for length) then you would also need to know the output of the previous block, since it gets added to the output of the current block.