That article fails to point out that rolling shutter effects only apply to current CMOS camera sensors - while CCD camera sensors use a global shutter, so that effect doesn't occur on them.
So if you want to avoid that issue entirely, just make sure to use a camera with CCD sensors, not CMOS sensors. My Canon camera uses CCD and all my movies with moving props yield only the blur one would expect of a non-zero exposure time.
I have no idea. Also not sure what resolution qualifies as HD.
But for reference, I bought a Canon PowerShot ELPH 115 IS (CCD) for about $120. (I think the ELPH 150 IS is the latest of that series.) I got it mostly because it can fit in a pocket and it is possible to shoot videos and still pictures with one hand. It is pretty much dedicated to the flight bag. I'm a novice at cameras so I'll let those who know the field give proper recommendations.
For some idea of what a moving prop looks like with video made with a CCD camera, here is a boring and shaky video (some resolution lost when uploaded) shot by me with the Canon while simultaneously taxiing over rough ground (note the prop isn't distorted; I also gave the auto-focus a work-out too):