When computing the hash, do you need anything after the number of difficulty bits?
The first 00000000 is for the nonce. The nonce is changed in order to get a different block header without reselecting transactions. It is what is used primarily for finding a valid block hash.

The long string of bytes beginning with 0000008 is SHA-256 padding. It is not actually part of the block header and whatever is doing the SHA-256 hashing should figure out and append this padding for you. It is part of the pre-processing step for SHA-256.