Wood suitable for routing
The first part of your question was addressed well over on woodworking.SE, at this link.

For the second part, there are a few reasons that I might not want to do a decorative detail on a piece of wood- 1. The wood and/or project isn't worth the effort, 2. The wood is hard on tools, or 3. The wood is hard on the wood worker.

Routing decorative detail can be a time-consuming process. I might need to create jigs or fixtures to guide the router. I might have to make a profile with multiple bits, each requiring set-up time. If the wood is especially hard, I might have to make a cut in multiple passes, sneaking up on the desired depth a little at a time.

Regarding #2- some woods (Teak, Cumaru e.g.) contain silica, and will quickly dull router bits that are used on them. I might limit the profile carved, or use an old bit that I don't mind sacrificing.

Regarding #3- some woods produce dust that can cause allergic reactions, and others (pressure treated, e.g.) contain chemicals that are harmful to people without proper protection.