Because Mythos of Brokkos doesn’t target the permanent cards in your graveyard, you may choose the card you searched for.

A permanent card is an artifact, creature, enchantment, land, or planeswalker card.

The abilities of the Mythos check what colors of mana were spent to cast the spell. It’s not an alternative cost to cast the spell.

If an effect copies the Mythos spell, no mana was spent to cast the copy, so the copy won’t receive the bonus.

The ability checks what mana was actually spent to cast a spell. If an effect allows you to spend mana “as though it were mana” of any color or type, that allows you to spend mana you couldn’t otherwise spend, but it doesn’t change what mana you spent to cast the spell.

If an effect allows you to cast a spell without paying its mana cost, you can’t choose to cast it and pay unless another rule or effect allows you to cast that spell for a cost. Similarly, you can’t waive a cost reduction unless that effect says you may.