My priorities are mechanical fitness, good looks, and availability/sustainability.
By mechanical fitness I mean wood that meets these criteria:
1) free from defects (knots, twist, runout, etc) unless used in a purely cosmetic manner.
2) doesn't possess any seriously bad characteristics (extremely oily, extremely unstable dimensionally with respect to moisture content, etc)
3) is stiff enough for the desired application
4) is light enough that the bass won't be unbalanced or too heavy
The vast majority of hardwoods and some softwoods meet those criteria.
After that, the next most important thing is that it looks good. Sometimes you want a plain wood to offset another piece of the instrument made from something more fantastic, or to make for a clean design. Other times you want lots of figure or color. Choice must match, i.e. maple, walnut, and cherry look awesome together but bubinga, padauk, and purpleheart clash with each other.
Finally, sustainability and availability. The more I build, the more I try to think carefully about where the wood lived when it was a tree, and how it got to me. I like buying wood from "the slab guy" I met on facebook, since his lumber is all reclaimed local stuff so there's essentially no impact to living trees. Whether this is legitimate or not is up for debate I suppose, but I feel better about that than about buying a piece of something exotic from a rando ebay supplier where you have no idea how it was harvested. When I do buy exotics, I try to at least buy them from vendors who have a reputation on the line.
"Tonewood" is not a thing for me. Any impact that wood selection has on tone is covered under mechanical properties, as far as I'm concerned. Build with wood that has the right properties and you'll get a good result. There's no need to buy wood that has traditionally been used for instruments, versus similar wood that hasn't.
Regarding your question about wood choices working well with each other. The way that can bite you is if you choose species that have vastly different expansion properties. Most hardwoods are fairly similar but if you end up with something that moves a lot glued to something that doesn't move much, you'll be more likely to have problems than if all your wood behaves similarly. Grain orientation among laminated parts makes a big difference there since most species have more movement in one dimension vs others. You can look a lot of these properties up online if you're curious.