Why is the voltage between 2 phases in the wye connection sqrt(3)*E in a delta-wye system
You need to understand phasor diagrams and then do some basic trigonometry. Consider this: -



The blue lines represent the voltages across each delta coil in your question. Now if you rearranged to a Y connection, to get the same voltage across each coil you would have to increase the line voltage (Vca, Vab or Vbc) by \$\sqrt3\$ to maintain the voltage across each coil as the same when it was delta wired. It's just trigonometry.