I've read somewhere (can't find source) that every freeze frame from an HD video is essentially a high-resolution photo (is this even true?).
No, not a scrap of truth unfortunately :( Here's why:
First up, it's not high resolution. Even 1080P ("Full HD") video, which not all cameras can do, is only 2 megapixels. That's a fraction of the sensor's resolution.
Secondly, video uses a much higher compression rate than still JPEG images, because video is bandwidth-intensive. Additionally, in most video formats (including AVCHD, H.264/AVC, various MPEG/DivX formats, but not "Motion JPEG"), most video frames use "inter-frame compression" which means most frames are not "key frames" and less information about them is stored. Inspection of individual frames will reveal they're lower quality where there is motion. However this looks normal when viewed as part of the video.
In video, a certain amount of motion blur is desireable, but in stills you would not want nearly as much. 1/48 or 1/60 second is normal for 24P and 30P in order to have a natural feeling motion blur (the effect is called "shutter angle" after old movie cameras). It is however possible for modern video to be shut at higher shutter speeds though.
Digital sensors can't provide such good quality image when they are in "Live view" mode, where there is no shutter but instead they're continuously fetching frames from the sensor. In CCDs this manifests as vertical streaking; in CMOS there are other issues (one is waviness during camera motion). The shutter that operates when taking stills enables the sensor to use a different mode where it doesn't have to reset so quickly and it can take a high quality image.
Video often has additional filtering applied than a still would, which loses some detail. One major one is interlacing, which effectively throws away a lot of vertical resolution. However, not many stills cameras with a video function would support interlacing. However, a lot of video will have a software antialiasing filter applied and maybe even a low pass filter, which soften the image but can help with compression. Depending on camera, video may also have more sharpening applied although nowadays with HD this practice is not so prevalent/detrimental.
What are the challenges, pros, and cons of selecting the best frame from a video as a way of taking still photos where timing is crucial?
Unless you had no other method whatsoever, I wouldn't recommend it. In fact I probably still wouldn't recommend it even if it was your only option. But if you did go ahead with this then make sure you can do 1080P, set the shutter speed to fast enough as if you were shooting still, and use Motion JPEG format (not AVC or any of the other formats I mentioned above). And realise that you're getting basically 2 megapixels.