Dream-Quest

Can I cast this on my Avatar, since they are a spellcaster?

Your avatar can't be disabled per their characteristics. Thus, they won't be able to go to sleep, and therefore won't be able to wake up. If you cast this on your Avatar, it will have no effect.

Using Dream-Quest, can I put to sleep one of my non-minion Spellcasters (e.g. Omphalos, River of Flame) or a spellcaster with restrictions (e.g. Earl of the Ivory Towers, Wind Sylph)?

Yes! An Omphalos will be disabled and thus not draw a spell that turn, but it cannot be hurt unless it's animated. Assuming it is not hurt, you may wake it up at the start of your next turn. For a site (e.g. River of Flame), it provides no Threshold while disabled, but it will wake up in time to produce mana next turn, due to the order of resolution in the .