You have two clues as to what is going on here first is the wood starting to rot or separate and the springs have rusted away. You have a leak of some type. Generally this is a mechanical valve at the bottom of the dishwasher that flushes water or recycles it.

If the door latches, but falls when opened, the springs at the bottom are probably broken. Usually you can see them if the bottom panel is removed. The flaking of the underside of the counter could by from the steam coming from the door vent. When the machine is working the hot steam condenses on the underside of the counter. If it hasn't swollen too badly, you can either coat the underside with a wtaerproof paint, polyurothane, or glue a thin piece of metal to deflect the steam up and over the counter top. I hope that helps!