For everyone's info, and because it could help someone else, here is the actual answer in my case.
While I had connected the white wire coming out of the AFCI to the ground bar, I hadn't connected the neutral from the circuit to the AFCI. In my case it was because I didn't realize the extra screw terminal was actually there and the documentation wasn't helpful.
Bottom line: If you install a combination AFCI and connect the hot but not the neutral, it will not pop until there is a load. After the fact, that actually makes sense. This is like a ground fault from the AFCI's point of view because it can't see the current going back through its neutral port.
No, Code says nowhere that a receptacle-type AFCI or GFCI is required; breaker-type GFCIs and AFCIs are also usable whenever AFCI or GFCI protection is called for by the NEC or otherwise desirable. In fact, circuit breaker manufacturers now offer a device that's both a GFCI and a CAFCI in the same package -- look for a DFCI (Dual-Function Circuit Interruptor) breaker.
As to your clarified question (cites from the 2014 NEC):
The refrigerator receptacle and the dedicated microwave receptacle do not require GFCI protection, as those receptacles are not installed to serve the kitchen countertop surfaces, and likely are more than 6' from the edge of your kitchen sink as well.
The DW circuit, however, does require a GFCI, even though it is hardwired, as per 210.8(D):
(D) Kitchen Dishwasher Branch Circuit. GFCI protection shall be
provided for outlets that supply dishwashers installed in dwelling unit
locations.
Note the word "outlets" in this passage, vs. the use of the term "receptacles" in 210.8(A) -- in Code parlance, an "outlet" is a place where power is tapped from a circuit to serve a utilization device, whether it be a hardwired device, a luminaire, or a cord-and-plug connected device, while a "receptacle" is what you plug cords into.
The disposal does not require GFCI protection either, unless its receptacle is within six feet of the top inside edge of the sink bowl "as the cord flies". (In other words, the receptacle, while not covered by 210.8(A) point 6, may fall under 210.8(A) point 7.)
Best Answer
It's pretty much all dependent on wattage
For the five kitchen appliances (and one lighting system) described in your post, the answer boils down entirely to wattage/amperage draw, as there are no specific requirements in the NEC for any of them to be on dedicated circuits; the fridge could be on a small appliance (kitchen countertop receptacle) branch circuit, and the remaining appliances can be on individual circuits or combined together as wattage permits; likewise, while the undercabinet lights can't be on a kitchen countertop circuit, they can go on any general lighting and appliance branch circuit that has the wattage to spare.
However, there are some constraints. Dishwashers, disposals, and microwaves all tend to be power-hungry beasts, meaning that you're going to have a hard time fitting even two of the three on the same branch circuit, and putting all three together on a single circuit is a non-starter. The fridge is the other constraint; if it's not on a kitchen countertop branch circuit, it must be on its own circuit, not a general lighting circuit. Given all this, I would look at pairing the dishwasher and disposal, if feasible, and also at putting the range hood and the undercabinet lights on the same circuit, given that you don't need a colossal commercial-style range hood to deal with domestic cooking needs, even if you have a jumbo range. (Commercial hoods are as beastly as they are because they have to keep air pollutants within OSHA limits throughout a restaurant cook's eight-hour shift despite near-continuous use of the appliance beneath them.) The microwave and fridge then wind up on their own circuits.
As to wiring methods, while there's nothing in Code that depends on that factor, the judicious use of individual THHN wires in ENT conduit ("smurf tube") can provide a major upgradeability benefit down the road. This is particularly handy in hard-to-upgrade spots, like when feeding the dishwasher and disposal, or for kitchen islands for that matter. You'll also have to make sure that you don't violate the 80% maximum total load rule for fixed cord-and-plug loads and the 50% maximum total load rule for fixed hardwired loads on convenience circuits (i.e. circuits that serve receptacles and/or lighting).