Q:

Does Kshatriya Dharma allow the sacking and pillaging of a country after it has been conquered, or the slaughter of its citizens during a conquest?

Does Kshatriya Dharma allow the sacking and pillaging of a capital after it has been conquered?
Historically, the armies of the Mlecchas, Asuras, and Rakshasas after defeating the army, killing the king, and conquering a city, would kill all the civilian men, rape all the women, make their children their slaves, loot all the property and wealth, and burn the remains of the city. 
Is this allowed as per Kshatriya Dharma (Arya Dharma)?

A:

Does Kshatriya Dharma allow the sacking and pillaging of a country after it has been conquered, or the slaughter of its citizens during a conquest?

No. At least not necessarily in the manner done by Mlecchas and demons, but in a different way.
According to the Manusmriti and the Mitakshara, when an army is sieging an enemy city, it is allowed to harass the inhabitants of that city or fort:

7.195 - After having besieged the foe, he shall halt, and proceed to harass his kingdom and continually vitiate his supply of fodder, food, water and fuel.
The siege has to be laid in such a manner that no one is allowed to enter nor any one allowed to get out
‘Kingdom’—i.e. territories outside the fortress occupied by the enemy.—This shall be ‘harassed’;—by kidnapping the inhabitants and persecuting them in various ways.
The ‘vitiating’ of fodder [food for domestic animals] etc. consists in spoiling them by mixing undesirable things with them.
This verse is quoted in Mitākṣarā (on 1.342), in support of the view that before a country has been entirely subjected, the conqueror should do nothing for the sake of the people of that country;
7.196 - He shall destroy the tanks, as also walls and ditches; he shall assail the enemy and shall frighten him—during the night.

All these, especially the siege, have the effect of starving and disturbing the civilians of that country, and appears to be allowed by scripture.
However, the outright massacre of non-combatants is strictly prohibited according to the law of the Aryas.
Here are the rules of engagement according to Hindu scripture:

Manu 7.91 - He shall not strike one who is standing on the ground, nor one who is a eunuch, nor the supplicant with joined palms [one who has surrendered], nor one with loosened hair, nor one who is seated, nor one who says ‘I am yours;’
Gautama (10.18).—‘No sin is incurred in slaying foes in
battle,—excepting those who have lost their horses, charioteers or
arms, those who join their hands in supplication, those who flee with
flying hair, those who sit. down with averted faces, those who have
climbed in flight on eminences or trees, messengers, and those who
declare themselves to be cows, or Brāhmaṇas.’
Baudhāyana (1.18.11).—‘He shall not light those in fear, intoxicated,
insane or out of their minds; nor those who have lost their armour;
nor with women, infants, aged men and Brāhmaṇas.’
Āpastamba (2.10, 11).—‘The Āryas forbid the killing of those who have
laid down their arms, of those who beg for mercy with flying hair or
joined palms, and of fugitives.’
Yajñavalkya (1.325).—‘He shall mot strike one who says I am yours, or
who is terrified, or deprived of arms, or who is engaged with another
person, or who has turned hack from the fight, or one who is only
looking on the battle.’
Devala (Vīramitrodaya-Rājanīti., p. 407).—‘Who is eating straw, who is
engaged with another, who is not actively engaged in fight, who is
seeking shelter, the imbecile, one pressed by another, the religious
student, the aged man, the outcast, the infant,—these shall not ho
struck in lawful battle, by soldiers, even in times of distress.’
Manu 7.92 - Nor one who is sleeping, nor him who is without his armour, nor one who is naked, nor one deprived of his weapons, nor one who is only looking on and not fighting, nor one who is engaged in fighting with ahother person;
Manu 7.93 - Nor one who has fallen in difficulties regarding weapons; nor one in distress, nor one severely wounded, nor one who is frightened, nor one who has turned back;—the king remembering the duties of honourable men.

And as a historical fact, Hindu armies rarely engaged in the indiscriminate slaughter of civilians like the armies of the Mlecchas, Asuras, and Rakshasas do.
From famous British historian Arthur L. Basham, in The Wonder that was India:

No other ancient lawgiver proclaimed such noble ideals of fair play in battle as did
Manu (p. 127). In all her history of warfare Hindu India has few
tales to tell of cities put to the sword or of the massacre of non-combatants. The ghastly sadism of the kings of Assyria, who
flayed their captives alive, is completely without parallel in ancient
India. There was sporadic cruelty and oppression no doubt, but, in
comparison with conditions in other early cultures, it was mild. To us
the most striking feature of ancient Indian civilization is its humanity.

As for the "sacking and pillaging" part, this is also strictly prohibited because it is stealing. Soldiers are not allowed to raid and loot peoples' homes, shops, agricultural fields, etc. They can only tax them lawfully.

