1) looks fine. since the directrix is horizontal the axis of symmetry is vertical, through the vertex.
2) Haven't looked at it.
3) You have a slip somewhere.
x = 4/5 x' + 3/5 y'
y= -3/5 x' + 4/5 y'
I lazily subbed these into 145x^2 + 120 xy + 180y^2 =900 using a CAS and it simplified to 100x'^2 + 225 y'^2 = 900.
This can be further simplified to (x'/3)^2 + (y'/2)^2 = 1.