Just as some dishwashers have soft food disposers in them (i.e., KA) you wouldn't put lemon wedges from glasses of ice tea or chicken and potato skins from dinner in the dishwasher, would you?Many people DO put such food scraps in their dishwasher, *because* it has a food disposer, even though they're not intended to take large quantities of such material. As for "disposable" plastics ... I routinely use butter-bowls and such for leftovers, and have never had an "incident" with them in my dishwasher (other than flipping and left filled with water/debris). The only damage I've had to plastic was a Mr. Coffee teamaker pitcher that warped in the bottom rack of a Frigidaire dishwasher that had a nuclear-powered, exposed heating element, LOL. I should have known better, but at the time I was used to my parents KitchenAid that had a non-exposed heating element with fan-forced drying and overlooked the difference.

I have never seen a DW or disposal with a cord and plug, but I don't get out much, if they sell the cords then I'm sure it is a fine way to go. One typical way is to run a 12/3 to a switch near the sink. From the switch the black circuit goes to the switch and then to the disposal, the red goes straight to the DW (or black to DW & red to disposal). Two separate 12/2 is also common, I prefer separate wires as the "loose neutral" problem can't happen that way. However it uses more copper and since there are only two multiwire neutral connections odds are you would not get a loose neutral problem, and it would be easy to find if you did. You could run the 14/3 to the disposal outlet and then run a 14/2 from the outlet through the wall to hardwire the dishwasher. Then the dishwasher will have no plug and you will have one less wire under your sink.

Per NEC 210.23(A)(2) You cannot add lighting to either appliance circuit if the rating of total utilization equipment fastened in place is greater than 50% of the circuit rating. That will probably rule out the 20A dishwasher circuit if typical @12A rating. But the disposal should be rated well less than 10+ amps. If so, then put the lighting on the 20A disposal circuit.

The NEC is the National Electric Code, but jurisdictions can adopt it in while, in part, or change any part they want for local use (depending on what authority they have, especially in Dillon Rule states). The small appliance branch circuits under the NEC are restricted to the wall-counter space receptacles in the kitchen, and with just a few exceptions (gas range, refrigerator if you really want to (IIRC)) but NO other outlets are allowed. No garbage disposals, dishwashers, lights, receptacles NOT on the wall-counter space. It is easy enough to overload the circuits with counter appliances that ARE allowed without allowing them to go anywhere else. Coffee makers, toasters, toaster ovens, and microwaves ca overload ONE of the circuits easily enough to cause nuisance tripping. By making sure appliances commonly used together are on the two separate circuits a lot of nuisance tripping can be eliminated. It can still be a problem if you try to use the microwave with a large kitchen mixer on the same circuit. If the two appliances are both started at the same time it may trip (both pull more power ant initial start up). Many kitchen appliances are very close to 15 amps (1800 watts) and at least part of the time may be over that briefly. Put two of them on the same 20 amp circuit and a nuisance trip is possible. The fact that many also cycle on and off in operation (microwave on less than 100% output, coffee makers pot heater, taster oven on temperature setting below full blast) and you may go weeks with little problem, and then have it happen almost daily, and then even stop again. If you have a lot of higher power kitchen appliances that may get used together you could even consider more than the minimum required two circuits.