Is the neutral a current-carrying conductor?

It depends on which of the three printed branches of NEC 310.15(E) describes your circuit, and 310.15(E) prints no general rule beyond those three: its own lead-in says neutral conductors are considered current carrying "in accordance with any of the following". Under (E)(1) a neutral that carries only the unbalanced current from the other conductors of the same circuit is not required to be counted. Under (E)(2) the neutral of a 3-wire circuit taken from a 4-wire 3-phase wye system is counted, because it carries approximately the same current as the line-to-neutral load currents of the other conductors. Under (E)(3) the neutral of a 4-wire 3-phase wye circuit is counted where the major portion of the load is nonlinear. The equipment grounding conductor is not counted when applying the adjustment, under 310.15(F).

The count is the total number of conductors in the raceway or cable, which is the Table 310.15(C)(1) footnote's own wording, so every circuit sharing the conduit is added together rather than counted separately. Pulled but unterminated spare conductors are included by that same footnote. Conductors connected to components that cannot be simultaneously energized, such as conductors on a mechanical interlock or a transfer switch, are excluded by it. Grounding and bonding conductors are excluded by 310.15(F). Then 310.15(E) decides, per circuit, whether that circuit's neutral is among them.

No, and the reason is the part that gets misread. A 120/240 V three-wire circuit is supplied from a center-tapped single-phase transformer, not a wye, so NEC 310.15(E)(2) does not reach it: (E)(2) is scoped by its own printed words to "a 4-wire, 3-phase, wye-connected system". That neutral carries the difference between the two legs, which is the unbalanced current (E)(1) describes, so it is not required to be counted. A three-wire circuit of two hots and a neutral is therefore two current-carrying conductors. Harmonics do load a single-phase three-wire neutral in the real world, and (E)(3) is scoped to a 4-wire 3-phase wye circuit and does not reach that case either, so the code text as printed does not require counting it. Your AHJ may.

Not on a single circuit taken from a wye. Two phase conductors plus a counted neutral is three current-carrying conductors, and Table 310.15(C)(1)'s own title is "Adjustment Factors for More Than Three Current-Carrying Conductors", so three or fewer is the absence of an adjustment rather than a row of the table. So a single (E)(2) circuit lands in 1-3, this tool's own no-adjustment key and not a row of the table. It starts to matter as soon as anything else is in the raceway, and a spare conductor is enough, because the count is the total in the raceway and spares are counted. A technician who learns "(E)(2) means count the neutral" and stops there will not understand why nothing changed.

The whole adjustment. Three phase conductors with a linear load is three current-carrying conductors, which lands in 1-3. The same physical install with the major portion of the load nonlinear counts the neutral under (E)(3), which makes it four, landing in 4-6. That is the first band the table prints an adjustment for, so the identical conduit is derated differently on the strength of what is plugged into it. Nonlinear load is switch-mode power supplies, LED drivers, variable-frequency drives, electronic ballasts, UPS and computing equipment.

310.15(E)(3) turns on the phrase "the major portion of the load" and does not quantify it, and no percentage appears in 310.15(E) or in the Table 310.15(C)(1) footnote, which is the text this tool holds under receipt. No number can be quoted for it, and this tool supplies none of its own. That judgement is yours and your AHJ's. If you cannot tell yet, answering that the load is mostly nonlinear counts the neutral, which raises the conductor count and so applies a smaller adjustment factor, and of the two answers it is the only one that cannot overstate what your conductor may carry.

Yes. The Table 310.15(C)(1) footnote counts "the total number of conductors in the raceway or cable, including spare conductors", so a pulled but unterminated spare is in the count. Leaving spares out lowers the count, which raises the adjustment factor, which permits more ampacity on a conductor that is sharing the conduit with them. That is the direction that matters.

No. NEC 310.15(F) excludes a grounding or bonding conductor when applying the provisions of 310.15(C)(1), and it applies whichever way 310.15(E) decides the neutral. Isolated and insulated grounding conductors are excluded on the same basis. Five wires in a conduit can be four current-carrying conductors plus one equipment ground.