The people at your lumber yard should be able to help you with your wood choice. I will just add a bit more (politely disagree) with some of what was said.
Mahogany is no more water resistant than many other hardwoods, and would need to be protected with a varnish, laquer or paint. Typically only woods with very high oil content in them naturally, are resistant to moisture. Those include cedar, teak and more exotic woods like cocobolo.
Also the strand orientation or "grain" of the wood, is not a very good indicator of whether or not the wood will twist. Although some woods are very prone to twisting and warping, it has more to do with the woods thickness, density, how it is cut (quartersawn vs. straightsawn vs. rotary cut), how it is dried, and as stated, the moisture content. Like iceblue said, wet wood is prone to twist when it dries, so buy dry stuff.
One other thing, balticbirch plywood is not the best stuff to stain, the pores are small in birch so use a pigment based stain if you do want to stain it. It will paint real nice though...