There are probably at least two things going on here that are contributing to the artifacts you see in your photos:

• Moire. When you take a picture of a regular pattern (like the rows and columns of a large LED display screen) the pattern you are photographing and the pattern on your camera's digital sensor can interfere with one another. It's like looking through two screen doors, one in front of the other, at the same time. The way the two patterns overlap when lined up creates the odd patterns we call moire.

Before you throw your DSLR away, be sure the pattern is showing up when you view your images at full resolution (i.e. at 100% where one screen pixel equals one pixel in the photo). Often moire can be caused by the scaling errors of reducing a high resolution image with repeating patterns in it to a lower resolution image for viewing. If you are viewing your 5D Mark II's 5616 x 3744 21MP images on the camera's 920K dot (just less than 1MP) rear LCD or on a typical computer monitor then the images are being scaled down significantly. Check a small area of the on-stage screen by pixel peeping it at 100% and see of the pattern is still there.

• Flicker/Modulation. LEDs work by flickering on and off very rapidly. Large display screens that use LED lights control the perceived brightness of each color of each pixel by modulating how long each LED bulb is on and how long it is off. This modulation is done at a fairly high frequency, but when you are shooting at a shutter time that is faster/shorter than one complete cycle of the modulation you will catch the screen with some of the lights pulsed on for the entire exposure, some pulsed off for the entire exposure, and some that pulsed on or off while the shutter was open. This can also create strange patterns that look a lot like moire.

The solution to flicker/modulation, as you have discovered, is to lengthen the shutter time until it is slow enough to average out the on/off pulses of each LED bulb in the display on stage. Of course that also means your rapidly moving subjects are now blurry from their motion.

Moire, and to a lesser extent pulse width modulation, can be dealt with in postprocessing with varying degrees of success. Doing selective sharpening by using a brush to select and reduce the sharpening of the LCD display in the photo can help smooth it out. You can control the sharpening individually for each viewing size/resolution and create, optimize, and export different resolutions of the image for different display environments. This can also help to reduce moire caused by scaling errors.

With regard to the effect of using flash:

• If the flash is being used in E-TTL mode with the camera in Av exposure mode or with Auto ISO selected the overall exposure of the ambient light might be getting reduced by the camera so that the influence of the light emitted by the LEDs in the screen is less in terms of the overall light in the photo.
• Since the sync speed is 1/200 the shutter time is also longer than when shooting at 1/320 or 1/400 without the flash. The rest of the subjects onstage are being "frozen" by the short duration of the flash. But the LED screen on the stage is emitting its own light for the entire time the shutter is open.
• The light from the flash may be getting into the screen and bouncing around inside and then reflecting back to the camera which would fill in the dark gaps between each bulb and smooth out the moire from the pattern of the screen's bulbs.