Left and right of sink. No other devices allowed. (No disposal.) Multiple outlet allowed as long as they are all on the same side of the sink. These MUST be GFCI.
Not quite. You need at least 1 outlet on each section of countertop (so yes, you need an outlet on either side of the sink.) However, they can be on the same circuit. You cannot have one circuit feeding all of the countertop outlets; you need a minimum of two circuits. But again, it doesn't matter if one circuit feeds outlets on both sides of the sink, and the other circuit feeds some other countertop outlets elsewhere in the kitchen.
Finally, these 2 circuits can ALSO feed other wall receptacles in the kitchen, pantry, breakfast nook/eating area, and the dining room. But they are to be used ONLY for receptacles (no hard-wired or permanently-installed appliances, and no lighting) and they are for receptacles ONLY in these rooms (not to be shared with receptacles in the living room or other rooms). This is the minimum required by NEC; of course you are welcome to split these receptacles up and feed them from separate circuits.
I am not an electrician or "NEC code lawyer". The last kitchen I wired, the dishwasher and the disposal shared the same 20A circuit. I put pigtails on both appliances and the duplex outlet was split. One "hot at all times" and one switched for the disposal. (I don't recall if I used a GFCI on this outlet or not, but it is a good idea. If you do a MWBC, you will nee two GFCI outlets.)
GFCI is required on dishwashers. Disposals- GFCI is not explicitly required in the NEC, but a good idea IMO.
You will get a lot of controversy over the rest.
Refrigerator - unless it is a "commercial" model, 15A is adequate and it can be shared with other appliance (maybe not the microwave).
Refrigerator can be on one of the SABCs (the two minimum circuits I talked about earlier), or you can put it on its own dedicated circuit.
Microwave/fan/light - 15A
20A is required by code. This is also required to be a dedicated circuit with nothing else on it.
Gas range - shared with microwave/fan/light (or maybe the refrigerator).
Can't be shared with microwave (see my comment above). But it can be on one of the SABCs.
Ceiling lights should be on a separate circuit.
This was a "budget" remodel and there was no money for a whole new load center. The house was old and very "oddly" wired. I needed four circuits, 2 counter top, 1 dishwasher/disposal and one refrigerator/microwave, plus the original ONE circuit that was there (yes, only ONE !) and of course the load center was full. Swapped in 4 tandems and I "made it work". Not inspected, but with THREE additional 20A circuit in that kitchen there was never an issue.