How to calculate chance of missing an error with overlapping hashes?
The usual way this is performed is through a Merkle tree, of which you will definitely find information on the Internet. What you can do is to separately hash each block (for performance reasons) and then hash the hashes together (in the correct order). Then you can simply store just the resulting hash.

In your scheme you perform a hash with half output. But you only want to trust the block after you've also verified the additional hash. So basically your "half-hashes" over the blocks are pretty useless. You may just as well hash the 4 blocks together or use a Merkle tree as described above.

Note that none of your constructions is likely to have more security than the output of a single hash (the state of both the single block hash and four-block hash would be the same after the first block), even though the output is larger than that. That said, I don't directly see how it could be less than the security of a single hash either.