So if you have a 1080P monitor watching a 720P video, there will not be a substantial quality loss unless you maximize the video to take up 1080 pixels. Since I have a 1200P monitor, I will see quality loss even on 1080P videos if I maximize it, although not quite as much because only 120 pixels are replicating colors where they originally wouldn't be, compared to 360 pixels enlarging a 720P video to a 1080P screen.
Now to the compression loss of quality. If you remember "The Wall Trailer" when it was first put up on SOSnewyork.com, it suffered from heavy compression quality losses, too much to even judge it's quality of work. Then when you look up the same video on YouTube from a channel like IGN, you will notice much better quality because there was less compression being done to lower the file size. Aside from the wall trailer though, there hasn't been any noticeable quality loss due to compression, atleast enough to skew your opinion on the videos. There will always be some form of quality loss when you compress it, but if its done right the loss is not in a substantial or noticeable amount as to skew your opinion.
If you have ever noticed with the comparisons of CD's to MP3's, the CD version will always be better, the same thing goes with BluRay versus standard DVD's. Why? Because less compressioning needs to be done to fit a movie or game on a Bluesy disc, and the same goes for CD's. Because of the less compressioning needed to be done, you can have better quality content. However for the sake of a 720P YouTube video, the compressioning will not usually be enough to change your opinion on say, a Crysis 2 video, as long as it isn't maximized.
Hope that helped!