Not exactly "Taboo" but some banking apps will refuse to load on Android if they detect your phone as rooted. Maybe there are apps doing similar stuff on iPhone as well, wouldn't surprise me.

Yes there's definitely a PR push which pretend that custom roms & rooted devices are insecure whereas security is an orthogonal concept to that in reality, there's a lot of custom roms which are more secure than the base rom.

Well, it typically voids your warranty entirely for one, and the terms of service usually use scary language telling you to never modify anything.

It can't void the warranty in the US or EU.