Getting the wrong hashes for bitcoin core packages
You are hashing the 404 page returned by bitcoin.org. You can see this by running:

$ curl -s https://bitcoin.org/bin/file-does-not-exist | sha256sum
69b00985c84e6633cc2e518914436ce55666cc10d1edf15ee442758d9f8d5219  -
$ curl -s https://bitcoin.org/bin/file-does-not-exist2 | sha256sum
69b00985c84e6633cc2e518914436ce55666cc10d1edf15ee442758d9f8d5219  -


You should instead run

$ curl -O https://bitcoin.org/bin/bitcoin-core-0.10.0/bitcoin-0.10.0.tar.gz
$ sha256sum bitcoin-0.10.0.tar.gz # shasum -a 256 would also work


This gives me

a516cf6d9f58a117607148405334b35d3178df1ba1c59229609d2bcd08d30624  bitcoin-0.10.0.tar.gz


which is the SHA256SUM listed in the associated SHA256SUMS.asc file.