Are existing non-tamper-resistant outlets grandfathered in NEC

Code requires two 20 ampere small appliance branch circuits, which it sounds like you have. There's no requirement that I'm aware of, that the receptacles near the sink have to be separate circuits. Though it could potentially be a local ammendment, so you'd have to chek with your local building department.

NEC also requires all countertop receptacles to be GFCI protected, not just those near the sink.

210.52(B)(2) is saying you should install 2 small appliance loads specifically for small appliances and are only allowed kitchens, dining rooms, and other specified locations. Usually they are circuits installed around the countertops of a kitchen. So you may not use them for general purpose receptacles in other rooms. They should not be used for disposals, dishwashers, etc.

Remember you can install more than two small appliance circuits if you see the need.

In your second comment you are misreading Table 210.24. It refers to circuits with a single receptacle. You can only connect no more than 10 receptacles on a 20A circuit or 8 receptacles on a 15A circuit. And all of that is for general power receptacles.

Remember you are installing small appliance receptacles. That means you may be connecting up an 1100W coffee maker and then say a 860W countertop Microwave and running them at the same time. Then you decide to pop on the toaster. If these are all on the same circuit, you be trying to get back to your panel to try and find the breaker you just tripped. And you haven't even had your first cup of coffee. Generally I would never put more than 4 duplex receptacles on a small appliance circuit.

Hope this helps.