GFCI Electrical – Is a GFCI Good for a Washer in a Utility Room?

We can divide this problem into two cases; namely, "instant trip" circuits and "device trip" circuits. (There are also "any device trips" circuits, but you do not have one of those.)

Circuits that trip a CAFCI instantly when the breaker is installed and switched ON without loads connected/switched on.

This is a clear sign of a wiring flaw in the circuit: either there is a parallel arc present due to faulty wiring (rare), or more likely, a ground fault is present in the circuit, also due to faulty wiring or a mis-wired circuit.

Examples of the latter include:

• landing the circuit neutral on the neutral bar instead of the breaker
• cross-connecting the circuit neutral to another circuit's neutral
• grounding the neutral at some point downstream of the main panel (this is most commonly the fault of bad subpanel installs, but can happen if someone does something wrong in a box as well)
• trying to put a single pole breaker on a multi-wire branch circuit or shared neutral run

Helpfully, you can glean diagnostic information from your breakers when troubleshooting this. As per this Square-D instruction sheet,

1. Turn the culprit AFCI OFF
2. Hold in the TEST button (white button) on the culprit AFCI
3. While holding in the TEST button, turn the culprit AFCI back on.
4. If the culprit AFCI trips instantly, it is sensing a ground fault (such as a shared or miswired neutral, although damaged wiring cannot be ruled out). If it delays a couple of seconds and then trips, it is sensing an arc fault (damaged wiring).

Circuits that trip a CAFCI when a specific device is switched on or plugged in

This is indicative of a faulty device -- in your case, a garbage disposal. Some older disposals use brushed "universal" motors, and these can produce more severe arcs if the brushes are worn, which could trick the AFCI into tripping. Having the motor rebuilt with new brushes would fix it in this case.

If your garbage disposal is a newer model using an AC induction motor, though, it is likely simply either a victim of water ingress, or just plain faulty; in either case, it's time to replace the whole disposal.

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.)