The only kinds of TVs that are prone to burn in are those with CRTs (picture tubes) and plasmas. It doesn't matter if the CRT TV is rear projection, front projection, or direct view; the issue is with the CRT, not the manner of projection. With your LCD projector, you could not burn in an image no matter how hard you tried. If you put a single image on the screen, turned the contrast and brightness all the way up, and left it on for a year, you would not burn in the image.
Strange things when switching sources is normal; do not worry about that.
As for the cables, Sony probably knows how to make them, and it is doubtful if any other cable would give you a better picture. Don't fall for magical cable thinking. Don't judge the performance by the appearance of the cable; judge it by its actual performance. You don't need thick cables. Of course, there is a minimum size that will work (which is fairly small, though how small depends on things like length), but there is no performance benefit to bigger than it needs to be. Very likely many of the "thicker" cables just have thicker insulation to impress fools who judge cables by looking at the cable rather than by measuring their performance.
So, if you play your games all you want, you will not hurt your TV, but do not play so much that you get carpal tunnel syndrome.
