Can area of a triangle be 0.5(c x a) instead of 0.5(a x c)?

[itex]ca = ac[/itex], so yes.

 

Not in that figure, unless angle B is 90 degrees.

 

using the magnitude of the cross-product of two vectors to find the ar

If you mean

Area of triangle = (1/2)||A x C|| = (1/2)||C x A||

as in, using the magnitude of the cross-product of two vectors to find the area of the triangle between them, then yes.

 

I didn't see the second part of your question. The direction of ||A x C|| or ||C x A|| doesn't matter because they have the same magnitude. That is why (1/2)||A x C|| = (1/2)||C x A|| is true.