What do ethical conduct (sila) and compassion have to do with the cessation of dukkha?

Namo Buddhaya. One very important aspect which has not been said in the answers so far is that cessation of Dukkha can not be achieved if the Kamma has not been exhausted.Suffering can not affect those who have done no bad deed. By developing loving kindness,compassion,altruistic joy and equanimity one can liberate the mind from past kamma. It leads to non-returning or Nirvana.

I quote the following sutta to support the answer:

“This noble disciple, bhikkhus, who is thus devoid of longing, devoid of ill will, unconfused, clearly comprehending, ever mindful, dwells pervading one quarter with a mind imbued with loving-kindness, likewise the second quarter, the third quarter, and the fourth quarter. Thus above, below, across, and everywhere, and to all as to himself, he dwells pervading the entire world with a mind imbued with loving-kindness, vast, exalted, measureless, without enmity, without ill will. He understands thus: ‘Previously, my mind was limited and undeveloped, but now it is measureless and well developed. No measurable kamma remains or persists there.’

“What do you think, bhikkhus, if a youth were to develop the liberation of mind by loving-kindness from his childhood on, would he do a bad deed?”

“No, Bhante.”

“Could suffering affect him if he does no bad deed?” “No, Bhante. For on what account could suffering affect one who does no bad deed?”

“A woman or a man should develop this liberation of mind by loving-kindness. A woman or a man cannot take this body with them when they go. Mortals have mind as their core.

“The noble disciple understands: ‘Whatever bad deed I did here in the past with this deed-born body is all to be experienced here. It will not follow along.’ When the liberation of mind by loving-kindness has been developed in this way, it leads to non-returning for a wise bhikkhu here who does not penetrate to a further liberation.

“This noble disciple, bhikkhus, who is thus devoid of longing, devoid of ill will, unconfused, clearly comprehending, ever mindful, dwells pervading one quarter with a mind imbued with compassion … with a mind imbued with altruistic joy … with a mind imbued with equanimity, likewise the second quarter, the third quarter, and the fourth quarter. Thus above, below, across, and everywhere, and to all as to himself, he dwells pervading the entire world with a mind imbued with equanimity, vast, exalted, measureless, without enmity, without ill will. He understands thus: ‘Previously, my mind was limited and undeveloped, but now it is measureless and well developed. No measurable kamma remains or persists there.’

“What do you think, bhikkhus, if a youth would develop the liberation of mind by equanimity, from his childhood on, would he do a bad deed?”

“No, Bhante.”

“Could suffering affect him if he does no bad deed?”

“No, Bhante. For on what account could suffering affect one who does no bad deed?”

“A woman or a man should develop this liberation of mind by equanimity. A woman or a man cannot take this body with them when they go. Mortals have mind as their core.

“The noble disciple understands: ‘Whatever bad deed I did here in the past with this deed-born body is all to be experienced here. It will not follow along.’ When the liberation of mind by equanimity has been developed in this way, it leads to non-returning for a wise bhikkhu here who does not penetrate to a further liberation.”

Sila(morality), Samadhi(concentration), Panna(wisdom) is the way to cessation of Dukkha. Sila disciplines bodily and verbal actions. Without that discipline you cannot discipline the mind, without disciplining the mind, you cannot cultivate wisdom.

There is a strong aspect in the training by the Buddha to achieve some self-analysis, some self-liberation.

But there is also another one, not often discussed but it occurs in some suttas (don't have the actual reference at the moment). This might be called a social one which introduces something like "reciprocity" or "echo":

If you've taken the vow that you won't kill - and the people around you can trust you, then the people around you need not fear getting killed, they can themselves feel more free and secure, and this is not only a good athmosphere for them but shall echo back to yourself and your freedom to move between them - and in general for the common social environment.

If you've taken the vow that you won't take what is not given - then the people around you need not fear that their things shall be taken away. When they know you are of that type you might walk through the fields (if it is a rural area) with much less things to be taken care of - so this again shall echo back to you - and in general for the common social environment (I really loved such a free- and trusting feeling when I wandered through the south of France, it was really new to me grown up in urban Germany...)

Think about the other vows - it should not be too difficult to image analoguous examples.

So, your keeping-the-silas and generating-trust-in-this improves an environment of freedom and - one shall even find out- of generosity and other good mental states and routines, thus reducing dukkha on both sides in such reciprocal process.

Kimattha Sutta (AN 11.1) explains the relationship of skillful virtues (sila) to Nibbana:

Ven. Ananda asked: “What is the purpose of skillful virtues? What is their reward?”

The Buddha's answer:

“Skillful virtues have freedom from remorse as their purpose, Ananda, and freedom from remorse as their reward.”

“And what is the purpose of freedom from remorse? What is its reward?”

“Freedom from remorse has joy as its purpose, joy as its reward.”

“And what is the purpose of joy? What is its reward?”

“Joy has rapture as its purpose, rapture as its reward.”

“And what is the purpose of rapture? What is its reward?”

“Rapture has serenity as its purpose, serenity as its reward.”

“And what is the purpose of serenity? What is its reward?”

“Serenity has pleasure as its purpose, pleasure as its reward.”

“And what is the purpose of pleasure? What is its reward?”

“Pleasure has concentration as its purpose, concentration as its reward.”

“And what is the purpose of concentration? What is its reward?”

“Concentration has knowledge & vision of things as they actually are as its purpose, knowledge & vision of things as they actually are as its reward.”

“And what is the purpose of knowledge & vision of things as they actually are? What is its reward?”

“Knowledge & vision of things as they actually are has disenchantment as its purpose, disenchantment as its reward.”

“And what is the purpose of disenchantment? What is its reward?”

“Disenchantment has dispassion as its purpose, dispassion as its reward.”

“And what is the purpose of dispassion? What is its reward?”

“Dispassion has knowledge & vision of release as its purpose, knowledge & vision of release as its reward.