What's the basis of these transcendental values?

The basis of transcendental values is that they complete things - transcendence is superior to nature because it is not confined by it, therefore it being based in transcendence means having greater inner freedom and more Being. Uniting the natural world with what transcends the natural world grants a quality of completeness, clarity and independence. Following transcendental values contributes to growth in this domain - to give an example, let us take a look at courage. Courage is a transcendental quality because it affirms the independence of the personality in the face of death, it affirms spirit over matter. It is not an evolutionarily useful quality at all, because it endangers life in order to benefit that which is more than life. Taking the materialist perspective on this issue ends up transforming courage into a kind of disability, hence the utterly ridiculous but popular belief in some circles that "men are expendable" since their only roles according to this Darwinist worldview is to inseminate and fight for the protection of others, as if men are idiotic automatons completely lacking in self-awareness.

To return to the more general point, these transcendent values can be affirmed in the form of commandments of God, but they can also be affirmed in most other spiritual worldviews, including for example that of the Ancient Greeks, which considered the world as a divine order and a cosmos which included both material and immaterial elements.

That's fundamental. All morality starts from life.

I can agree with this second statement, but with the reservations that not all forms of morality have an objective and meaningful character, and that this does not in itself imply biological life has primacy.

Our very sense of morality comes from evolutionary pressures that caused us to develop such feelings in order to preserve life.

This is not really true, because history has many examples of forms of morality that do not at all aim to preserve life. These examples can be both positive and negative. As to where morality comes from, that is a more difficult matter to address.

Morals such as honor or courage only have meaning in the context of human life. They hold no value in some barren wasteland like mars.

I partly agree with this, but your perspective is too materialistic. If our definition of human beings includes not only material but also spiritual aspects, it is perfectly sensible to think that by far the most objective meaning will be found - expressed in one form or another - in human life, with Mars having very, very little relevance in the grand scheme of things. This is a bit of a tricky subject, however, since the Anthropocentrist position is also wrong - from a Traditionalist perspective, the only thing that makes humans special is our much greater ability to participate in the divine order.

The issue is, is one of finding meaning and purpose in life.

I agree, but finding meaning in life does not necessarily mean deriving meaning from life, or at least, life conceived unilaterally to the exclusion of the spirit.

You can derive a sort of meaning from a darwinian view that you are the product of billions of years of evolution and it is your purpose in life to conform to adaptive behaviors.

I do not find this vision compelling because it's essentially an appeal to nature, but not all people naturally follow "adaptive behaviours". Many people are not capable of being "adaptive" at all, and some value other things higher than being "adaptive" - heroes, saints, philosophers etc. all prioritise other things above "adaptive behaviours". Personally, I also can't see the meaning in living your life according to a set of rules just because they perpetuate life. Materialistically, it makes more sense to focus on the individual life instead of "the continuation of the species". In fact, I think nature itself affirms this - some animals, like rabbits for example, occasionally kill their young.

I'll concede, this is a solid point. The issue again comes to why exactly are values such as generosity and altruism considered good or essential? How did these values come to be in the first place? They have become so because they were evolutionarily adaptive. A nation that abandons such values will have low trust and support amongst themselves and will collapse in the face of one that has not dropped these values.

I disagree. I have an alterative explanation on what makes these values important and why they are so widespread. While virtues are to an extent culturally constructed, I think that all authentic virtues have a transcendent character, hence why they are valued. Charity, for example, not only benefits social cohesion, but also develops detachment from material goods and the material world. Just like with the earlier example of courage, this increases the inner freedom of the human personality and makes men fearless. These qualities of independence and fearlessness are what helps moral societies triumph over immoral ones - not some sense of materialistic solidarity, which is ultimately the solidarity proper to criminals, traitors and thieves.

Unfortunately, this is the point where I become somewhat Reddit. The supernatural law is something that is vague and incomprehensible. You cannot establish a solid moral or intellectual foundation upon something you have no evidence or cannot even fully qualify. How do you know you have a soul? How do you know whether this soul goes to heaven, hell, Valhalla, purgatory, or Elysium after death? How does one measure the value of one soul against another?

You ask some very good questions, but I disagree that it is impossible to establish solid moral and intellectual structures on the basis of transcendent values - every Traditional culture has done this. The Greeks, the Romans, the civilisation of India - all of these are good examples of just that.

As to your questions, the Traditional conception is that you either dedicate your life to spiritual pursuits in order to experience spiritual reality yourself, or you have to take those who have experienced it on their word. Those are the only options. Just like the average person depends on an expert builder to build a house properly, so it is also with spirituality. Only people who have invested sufficient effort can claim to know anything about it. The rest have to trust them, or else give up on the pursuit entirely - be it spirituality, or construction, or dentistry. The two major issues today (in the "Kali Yuga") is that people have indeed given up on spirituality, and that the Traditional structures and institutions which preserved and transmitted spiritual knowledge have been destroyed. If we were having this conversation in an ancient Greek city, I would be able to refer you to a mystery cult where you would be able to experience spirituality for yourself, but such institutions no longer exist. Nevertheless, even today there are still some valid methods for getting answers for the questions you asked through direct spiritual experience - they are just harder and less accessible. There are also philosophical ways to rationally address your questions, but traditionally the normal way to address them is through spiritual experience.

such emphasis on souls and other forms of abstract, magical (I don't like to use this term but still) beliefs opens the door to ruinous egalitarianism...

What physical(material) differences...

I don't know if you are aware of this, but I actually had a fairly in-depth discussion with the user you mentioned on this matter. Here is his initial comment. You can find my attempt to refute him below. I will quote my most relevant arguments:

Doctrines that acknowledge the existence of the soul are radically, extremely inegalitarian, and we have historical evidence of this dating back to Plato, at the very least. No two souls are the same, and each differs from each on the basis of character, personality, dignity, wisdom, virtue, nature, affinities, capacity and lastly, content - much of which is received organically from a traditional society. Your demand for deciding on an exclusive preference for either body or soul is also completely unjustifiable and would have disastrous intellectual consequences for anyone who takes it seriously, because each of those is obviously important in its own way. Not to mention that rather than souls, your problem seems to be with the definition of human beings as interchangable, atomised and purely rational actors, which is a decidedly modern definition and has nothing to do at all with the traditional concept of the soul.

To summarise some of the other things we covered in that thread, we discussed Cartesianism and the modern re-definition of the soul as a purely rational faculty, which is a kind of blank slatism and is also completely antitraditional. In Traditional civilisations, the soul was not at all abstract but had very concrete features. I also rejected the claim that moral and spiritual qualities are unverifiable, which I consider completely self-evident and see no need in proving. Finally, I raised the point that ancient, European, pagan ethnoreligions still assigned primacy to the soul over the body, and yet this did not in any way prejudice their ability to function as ethnoreligions - consequently, the claim that assigning primacy to the spirit automatically presupposes some sort of all-encompassing egalitarianism must necessarily be a false one.