Yes, Omniscience will let you cast spells without paying their mana cost regardless of where they are, including the top card of your library with Future Sight out.
This is incorrect, and if you are referring to the old printed text of Future Sight (i.e., "...as though it were in your hand."), it never applied in that way. The "as though" text simply created a condition that was true for a specific thing. In that case, it would have been Future Sight's effect that allowed you to play the top card of your library, since normally you only had permission to play cards from your hand. It would not have been true for any other effect, especially for something like Omniscience...
609.4. Some effects state that a player may do something “as though” some condition were true or a creature can do something “as though” some condition were true. This applies only to the stated effect. For purposes of that effect, treat the game exactly as if the stated condition were true. For all other purposes, treat the game normally.
...and was important for things like suspend, as the permission to play the top card of your library doesn't actually make the card in the player's hand. (It's also why that language was removed, because it was misinterpreted to apply to situations it shouldn't have.)