If you cast Animate Library after Split Screen, Animate Library will be attached to only one of the four libraries. This is true even if you cast them in the other order.

If you also control Yet Another Aether Vortex, any of the four revealed cards that are permanent cards will also be on the battlefield.

If you cast a second Split Screen, one of your four libraries will itself become four libraries. You’ll end up with seven libraries. If one Split Screen leaves the battlefield, you’ll shuffle all your libraries together. The other Split Screen will still apply, so you’ll play with the top card of your one library revealed. When that one leaves the battlefield, you’ll shuffle your library.

If one of your libraries is empty, you won’t lose unless you try to draw from that one. Don’t do that. If all of your libraries are empty, you may not have much choice.

If you draw a card, you can draw from any of your four libraries.