When our wallets begin with a private key, do they translate to binary or do they start by hashing the hex or Base58 ASCII representation of the number?
It depends on what they want to do with it.
If the key is generated to be stored in the wallet file, some wallets convert it to base58 (called WIF) and store it in the file. The SHA256 hash is only for computation of the checksum.
If the wallet is supposed to use it to sign a transaction or compute the public key and the address, they'll have to convert it to a 256-bit integer which in most cases an array of unsigned 32-bit integers that would stored the bits for subsequent arithmetics. There is no hashing involved in ECC and only for checksum in base58 address in last step.