Help powering new light and switch off existing switch
Stop.  You can't just grab any old wire.  

You need two specific wires: (always) hot, and neutral.  And they need to come from the same cable.  This point is very important for not setting up a current loop, causing wire vibration, eddy current heating, and arc failures. 

Now, switches which have a white wire are no help.  Normal switches don't take neutral, so when you see a white wire going to a plain switch, that wire is not neutral (and should have been marked, but the installer shortcutted as usual).  So the cables going to those switches can be disregarded.   

I see two switches with no white wires, and that means they probably have a real neutral somewhere in the box.  You should see a wire-nutted bundle of all-white wires.  So you need to find that bundle, and follow those whites back to their cables.  Among the other wires in one of those cables, you should be able to find an always-hot. If not, you're out of luck, and will have to tap somewhere else.