First of all installing does more than speed up loading times. It reduces noise dramatically, in free roam games it reduces or eliminates pop in, it reduces amount of spinning parts in your xbox thus less heat.
And about safety, the moment you flashed your xbox you threw away safety. Nothing can guarantee that you will not be banned sooner or later. But yes you can install original games on HDD without problems, but using originals won't make you any safer.
C2D E6750 @ 3.2Ghz, 4GB 800MHz DDR2 4-4-4-12, GeForce GTX 260 c216 OC 896MB, 3.2TB, Windows 7 Ultimate x64
Xbox 360 Elite, PS2 Slim, Xbox