Quote from @PlayHearthstone
Does Yogg Saron's battlecry target friendly characters ?
Yes, Yogg-Saron's Battlecry affects all potential targets, including your own hero. (Source)
Are the spells randomly chosen from the spells you've cast? Or are they completely randomly picked from all spells?
Yogg-Saron's spells are randomly selected from all collectible spells, and one is cast for each spell you've played. (Source)
Will the spells Yogg-Saron casts be from any class, or from your own? And I assume Standard-only spells in Standard.
Random effects in Standard will only use cards eligible for Standard play, so Yogg-Saron would only cast Standard spells. (Source)
Can Yogg-Saron hit himself?
Yes, Yogg-Saron can target himself, but even if he's destroyed or transformed, the full Battlecry still takes place. (Source)
If a card provides options, such as Raven Idol or Wrath, Yogg-Saron randomly selects one of those options when the spell is cast. (Source)
What if Yogg-Saron uses a card that can't activate on anything? like if it plays backstab but all minions are damaged?
In a case where one of Yogg-Saron's spells has no eligible targets, it fizzles. (Source)
How random is "random target?" Can spells that normally only affect enemy minions target yours instead?
Yogg-Saron follows the rules, so the spells will follow their card text as though your hero cast them. (Source)
Random targets means my fireballs can hit my own minions and my buffs can land on the enemy?
That's correct! If a spell doesn't have text to prevent it, then any eligible target could be chosen. (Source)