Is it by law necessary the White House stay white, or could a President change it to meet his wife's preference?

Tie dye!...;-D...

I'd imagine it is.   As a national monument and treasure. 

I'd personally like to see it painted safety orange or something obnoxious like that.

Probably.  It is called White House because the English burned it during the war and some type white wash was used to cover the burn marks.  Roosevelt was the first to paint it with white paint and he officially proclaimed it to be the White House.  I believe one would play hell trying to paint it victorian blue, but wouldn't it be lovely............with crimson doors..