It matters little what you use as mass to shim the plywood; it just needs to be clean, dense and able to nail through. Shingles, either asphalt or wood undercourse are commonly used. Use the leveling feet on the stove for final adjustments when reinstalling that.

You're right, you need to get those cabinets really level. Here's what I can offer, based on DIY installing two kitchens: I don't know about face frame cabinets, but we DIY-ed the install of our frameless cabs and they definitely do twist and rack and get out of square if you don't level super carefully! They also may not be perfectly square from shipping. Our solution was a laser level and a whole lot of time and painstaking patience. Rather than using the easier leveling method of shooting a line and measuring back to it with a tape measure, we actually set the level up at the exact height to shoot a line just kissing the tops of the cabinets at the highest point. Then, one at a time, starting with the highest, we leveled them so that each corner was exactly just kissed by the laser line. I looked at the EZ Level system and it looks like it should make all the leveling issues easier. You will need to choose your starting point carefully. If you suspect that some of the cabinets may be truly taller or deeper than others, get out your measuring tape and check that. You will then use those pieces as the outside/highest/deepest starting point, since you can shim the others out or up to match those, but you don't want to be cutting the others down to match... Now put a laser level on the floor and shoot a line all around the room. Measure down with a tape measure and find the HIGHEST point of the floor, both along the back of the wall and 24" out. Your floor won't be perfectly level, so you need to find that point. Then measure up the height of the cabinet carcass, plus a little bit for wiggle room, and shoot a laser line along the wall or draw it on. That's your basic guide for the top edge. In a normal install, with shims, you then start with the cabinet that's at the highest point. Using the laser level you patiently shim each corner until it is perfectly level (and square, by shooting verticals). Then you do the next one. It looks like your EZ LEvel system will make that even easier, and allow lots of control. So I think if you measure the cabinets, find the maximum depth and width, and the high point of the floor, and work off those, you should be OK. Don't just start with the first cabinet and move along the row, though, that won't work. You need to start with the one that's at the highest point in the room, and/or is tallest or deepest. In cases where the cabinet needs to be installed a little bit away from the wall in order for the fronts to be flush, you will need to place some wooden shims between the cabinet and wall to make up the difference at the spots your fasteners go through. Good luck, I know its nerve wracking!

I ended up going with Showplace because the kitchen designer was really helpful and invested in improving on the architect's poor design and I felt that he would personally stand behind the product if we ran into any issues. Our cabinets are ready but will be delivered in a couple of weeks. However, if I already had a layout I was happy with or was just replacing cabinets in an existing kitchen, I think the Cabico cabinets would have been a better choice for me. It's all a blur now, but I preferred the color selection and overall they would have been more cost effective because I think they are cheaper to begin with and you could use a combination of custom and semi-custom cabinets from their line to meet your needs. With Showplace I found that every time I made what was a small change in the design from my perspective, it was incredibly expensive, and we were cutting corners to avoid completely blowing my budget. By cutting corners I mean not including cabinet accessories that I could get cheaper aftermarket or not putting in glass door cabinets that I had originally had in mind. Overall, I am happy with my concept and the turnaround on the cabinets is quick (about 6 weeks).

I'd do the opposite of what everyone is suggesting here. I'd put the sink under the beam and move the dishwasher to the left side. Then above the sink I'd do no cabinets at all. Then I'd extend all cabinets to the height above the fridge(42 or 36 or whatever it is you want). This has a few advantages. One, you get a much longer run of useful counter to the left of your sink. It also moves your utilitarian appliance away from the door. Even the prettiest dishwashers are.....dishwashers. You also can add some nice lighting above the sink, and it leaves some open wall space to do something flashy with backsplash tile. You also end up with no cabinets anywhere near the beam, keeping it from looking like any sort of interruption to the eye.