Pretty much any stable hardwood will be fine for both the neck and the fretboard, I agree that something meant for flooring will probably be fine.
I would skip the truss rod. Even with steel strings, there's really not much tension on a ukulele neck, and also - the neck is so short that there's not much "leverage" for the strings to pull it out of shape. I've built a number of steel stringed tenor guitars, ukuleles, and short scale 6 string guitars in roughly the size you're aiming for, and have never had a problem without a truss rod. I generally build them with typical hardwoods (maple, walnut, cherry) and make them dead flat. The minute amount of curve pulled in by the strings ends up just about right for the tiny amount of relief you want on a neck like this.
What are you planning for pickups? On my early electric steel stringed ukes I used a precision bass pickup, with each half positioned and wired separately as if it was a pair of single coil pickups. The precision pickup halves are just about perfect in terms of width and pole spacing for a uke. But these days there are a lot of pickups to choose from, meant for cigar box guitars and other smaller guitar applications. I've also used blade-style guitar pickups which work fine but they're a bit wide-looking...