Example... my opponent hit say, my minotaur unit in the flank. Could my doom-bull issue a challenge if he has a champion and swap places with my minotaur? (although technically he could just refuse)

A champion or character can only issue a challenge if they are in actual base to base contact with an opponent (any opponent). So, as I understand your description, the answer is 'no.'

You can maybe issue the challenge, but as there are no one to accept nor decline, it wouldn't have an effect. There are stuff that would let you do it anyway, like the dwarf oathstone, but otherwize, no, a challenge cannot take place...

To be eligable to accept or issue a challenge, a character must be engaged with a unit containing one or more characters/unit champs. And even then, only those units are eligible to accept/refuse. If the character is unengaged at the start of its next movement phase, it can then move to a fighting position and subsequently be involved in challenges. Generally speaking, its best to put strong combat characters on the corner of a unit for this very reason.