Wiring dishwasher and garbage disposal
The 12/3 and 20A is perfectly fine. This is a Multi-Wire Branch Circuit or MWBC. The basic idea is that by having two breakers on opposite legs (240V between them), you get 2 x 120V using 3 wires (hot/hot/neutral) instead of 4 wires (hot/hot/neutral/neutral). The catch is that the circuits have to turn off together at the breaker (your double breaker takes care of that).
From a safety standpoint, a disposal and dishwasher on 2 x 20A is really the same as if they shared a single 20A circuit. If one of the devices overloaded then the breaker would trip at the same 20A that it would trip if they were on the same circuit if the other device wasn't running at the same time. In reality, most overloads are going to be way over (short circuit or just a massive overload) and 15A vs. 20A just wouldn't make a difference anyway.
It also happens to be right now that you could put the two devices together on a single 20A circuit (sum &lt; 16A) but if you replaced one of them (e.g., installed a more powerful disposal) you could end up with a problem. 2 x 20A = guaranteed no problem.
As a general rule, you can't mix and match circuit breaker sizes, wire sizes and devices. For example, you can't put a 15A receptacle on a 30A circuit. However, 15A and 20A are a special situation in the NEC. The end result is that, generally speaking, most devices that go on a 15A circuit (15A receptacles, a lot of built-in appliances except for ovens &amp; cooktops) can just as easily go on a 20A circuit.
Just to be sure, I spot-checked a major US manufacturer dishwasher selected at random and the installation guide clearly states that it can be installed on a 15A or 20A circuit and plug-in or hardwired. Local codes may have specific requirements.
In this case, the one big catch is GFCI. You may or may not need GFCI on this circuit. If you do, then it gets a bit more complicated - either a GFCI double-breaker (expensive) or 2 separate GFCI receptacles instead of a duplex receptacle. Assuming GFCI is not needed, then depending on the instructions for your dishwasher and disposal, you can do one of the following:
Hardwired

12/3 from breaker to a junction box.
Short 12/3 to switch box above counter. Black hot to incoming black, red switched hot to disposal black hot, white neutral to incoming neutral and disposal neutral and dishwasher neutral.
Dishwasher black hot to incoming red, dishwasher white neutral to incoming neutral and disposal neutral and dishwasher neutral.

Receptacles

Use a 20A duplex. Normally a 15A duplex is OK on a 20A circuit, but that is when you are thereby getting 2 15A receptacles on one 20A breaker. In this case, each 1/2 has 20A, so I suspect (can't say 100%) that you must use duplex 20A. And nothing is lost by doing that (cost is same or perhaps 50 cents different).
12/3 from breaker to a junction box.
Short 12/3 to switch box above counter. Black hot to incoming black, red switched hot to top receptacle hot, white neutral to incoming neutral and receptacle neutral.
Incoming red to bottom receptacle hot.
Remove the hot-side tab from the duplex receptacle. Do not remove the neutral-side tab.
Plug disposal in top and dishwasher in bottom.

It is also possible to have one device hardwired and the other plug-in.