Yeah, I've done it a few times. I just make sure that their "planning and preparation" really does include planning possible diversion to one of the listed airports. Better to CYA and have them skip airports listed in the endorsement than have them stop for an "emergency" bathroom break at an airport they aren't signed off for cause they couldn't hold it anymore (happened to a CFI I know).
yup. I often include at least the bigger class D nearby and any airports we have been to in between the origin and destination... in many cases, its a re-authorization of airports I have endorsed previously.