How much of our initial conditions come from past Kamma?

These ideas arise from interpretations of the Cula-kammavibhanga Sutta. However this sutta is questionable both in terms of its composition (see this discussion) & visible actual reality. Or otherwise, it is wrongly interpreted due to the complex 'dhamma language' contained in it.

In the visible actual reality of the world, some of the most wealthy people do very evil things to acquire their wealth & some of the most beautiful people live lives of emotional turmoil. It is highly improbable such people did something exceptionally good in a past life & then became so evil & neurotic in the next life. Further, we can know exactly the steps a person look in this life to become wealthy, as written in their biographies (eg. Bill Gates). This shows their wealth is unrelated to any actions performed in a past life but often related to greed & deceit in the present time.

This visible reality is confirmed by suttas such as SN 1.28 & SN 3.7, which state:

Those of great wealth and property,

Even khattiyas who rule the country,

Look at each other with greedy eyes,

Insatiable in sensual pleasures.

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I saw that even affluent nobles, affluent brahmans & affluent householders — rich, with great wealth & property, with vast amounts of gold & silver, vast amounts of valuables & commodities, vast amounts of wealth & grain — tell deliberate lies with sensual pleasures as the cause, sensual pleasures as the reason, simply for the sake of sensual pleasures.

That's the way it is, great king! That's the way it is! Even affluent nobles, affluent brahmans, & affluent householders... tell deliberate lies with sensual pleasures as the cause, sensual pleasures as the reason, simply for the sake of sensual pleasures. That will lead to their long-term harm & pain.

That is what the Blessed One said. Having said that, the One Well-Gone, the Teacher, said further:

Impassioned with sensual possessions, greedy, dazed by sensual pleasures, they don't awaken to the fact that they've gone too far — like fish into a trap set out. Afterwards it's bitter for them: evil for them the result.

Or in the visible actual reality of the world, some individuals born into the world into very poor families eventually become very wealthy. It is illogical that a person with a disposition towards accruing wealth would choose to be reborn into a poor family.

Or in the visible actual reality of the world, it is rare to find beautiful children born to ugly parents. It would seem quite obvious that physical beauty is related to cellular genetics rather than to past lives.

MN 64 states when a new child comes into the world, it has underlying tendencies (anusaya) that are common to all new born children. MN 64 does not support the Cula-kammavibhanga Sutta & state new born children are born with underlying tendencies to be short-lived & long-lived people are to be seen, sickly & healthy, ugly & beautiful, uninfluential & influential, poor & rich, low-born & high-born, stupid & discerning, etc.

Similarly, AN 3.61 states whatever happiness, suffering or neither-happiness-nor-suffering a person experiences is not caused by what was done in the past. AN 3.61 states when one falls back on what was done in the past as being essential, there will be no desire & no effort to change one's life situation and one will abide bewildered & unprotected.

The Mallikadevi Sutta is very similar to the Cula-kammavibhanga Sutta however it uses more straightforward & temporal language. It does not use words such as 'kāyassa bhedā paraṃ maraṇā', which have specific real meanings but are capable of being interpreted in worldly ways. It seems obvious the Mallikadevi Sutta is referring to spiritual wealth & spiritual beauty rather than physical wealth & physical beauty because it states a woman getting angry cause her to reappear (paccājāyati) as ugly.

There are many suttas that distinguish between physical & spiritual attributes, such as:

There are these five kinds of wealth. What five? The wealth of faith, the wealth of virtuous behavior, the wealth of learning, the wealth of generosity & the wealth of wisdom. AN 5.47

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Householder, a noble disciple who gives food gives the recipients four things. What four? He gives life, beauty, happiness and strength. (1) Having given life, he partakes of life, whether celestial or human. (2) Having given beauty, he partakes of beauty, whether celestial or human. (3) Having given happiness, he partakes of happiness, whether celestial or human. (4) Having given strength, he partakes of strength, whether celestial or human. AN 4.58

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Dhammapada Verse 204: Health is the greatest gift, contentment is the greatest wealth, a trusted friend is the best relative, Nibbana is the greatest bliss.

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As the stainless moon

moving through the sphere of space

outshines with its radiance

all the stars in the world,

so one accomplished in virtuous behavior,

a person endowed with faith,

outshines by generosity

all the misers in the world.

the Perfectly Enlightened One’s disciple,

surpasses the miserly person

in five specific respects:

life span and glory,

beauty and happiness.

Possessed of wealth

he rejoices in heaven.

AN 5.31

A being's subconsciousness and most of the conditioning during a given alife is decided by the regenerative karma, which is past Karma.

Natural order (Niyama) of all being are only partly affected by Karma. Complex interference of the other factors may negate or even enhance some effects of Karma which some experiences may have nothing to do with Karma and based solely on one or more of the other factors.

A person with negative Karma has the possibility that this last thought not being good hence a possibility of a lower life. Karma is complex and only fully comprehended by a Buddha alone hence it is difficult to answer question on Karma with definiteness.