Most of the time, a dishwasher isn't flushed with the cabinet. So it's normal for it to stick out a little. Ideally, a dishwasher door sticks out 2.5″ from the cabinet frame, 1.5″ in front of the counter.
Therefore, the correct positioning is to set the front of the dishwasher – not the door – flush with the front of the cabinets.
The gap is necessary because dishwashers create a lot of heat and humidity. It can damage the counters to be exposed to these elements too closely. Most manufacturers recommend a gap of 1/2-3/4 of an inch.
Cover the joint between the panel and the existing cabinetry, if it is visible, using caulking or wood putty colored to match your cabinets. Place a bead of paintable caulking all along the seam and smooth it with your finger. Paint if necessary to create a seamless finish.
Corner locations need at least 2 inches of space between the side of the dishwasher door (when open) and the wall or cabinet. Check for any obstructions that would cause the dishwasher motor to touch the floor of the opening, such as carpeting or loose, uneven tile.
Many dishwashers have 1/2" - 3/4" gap or clearance to the counter top. This is not unusual. Many manufacturers require a certain amount of "clearance" to the top of the counter top/sides of the cabinets due to heat and humidity that comes of the machine. That's why I suggested a 1/2" plywood and nothing thicker.
No, integrated dishwashers do not need a cabinet. Instead, you need an opening where the dishwasher can be installed with sufficient space to accommodate the appliance's size and provide access to the kitchen's plumbing.
Standard Dishwashers
The vast majority of undercounter dishwashers measure approximately 24 inches in width, 24 inches in depth and 35 inches in height with adjustable feet. This means that the rough opening in your cabinets must be 24- by 24- by 35-inch.
A standard size dishwasher is made for a kitchen cabinet opening that is 24 inches wide, 24 inches deep and 35 inches high. Some dishwasher models have the same standard height and depth but come in different width sizes.
Considered the standard style, typical built-in dishwashers are about 24 inches wide, 24 inches deep, and 35 inches tall to fit most cabinet openings in residential kitchens.
An integrated dishwasher (also known as a built-in or built-under dishwasher) is designed to match the rest of your kitchen cabinetry. Integrated dishwashers fit beneath your counter top and provide a neat, flush finish to your existing decor.
Similar to your washing machine, a freestanding dishwasher can be placed almost anywhere there's a water connection and a drainage point. You do not need to build in a freestanding unit and it will come with sides and top cover so that you can safely place it wherever it will fit. Shop Freestanding Dishwashers.
The plywood has nothing to do with spanning the dishwasher gap: usually kitchen cabinets have no top, and the only support for the countertop is provided by the sides of the cabinets. Whether there is a dishwasher or a drawer underneath makes no difference.