Second Noble Truth

Noble truth of origination of stress is explained also in Tittha Sutta:

"And what is the noble truth of the origination of stress?

"From ignorance as a requisite condition come fabrications.
From fabrications as a requisite condition comes consciousness.
From consciousness as a requisite condition comes name-and-
form.
From name-and-form as a requisite condition come the six
sense media.
From the six sense media as a requisite condition comes contact.
From contact as a requisite condition comes feeling.
From feeling as a requisite condition comes craving.
From craving as a requisite condition comes clinging/sustenance.
From clinging/sustenance as a requisite condition comes
becoming.
From becoming as a requisite condition comes birth.
From birth as a requisite condition, then old age and death,
sorrow, lamentation, pain, distress, and despair come into play.
Such is the origination of this entire mass of stress and suffering.
"This is called the noble truth of the origination of stress.

Thus birth is the requisite condition comes pain, distress, despair etc...

Regarding the cases like of children they can be explained as diversity in Kamma and result of diversified kammas. From the Nibbedhika Sutta:

"'Kamma should be known. The cause by which kamma comes into play should be known. The diversity in kamma should be known. The result of kamma should be known. The cessation of kamma should be known. The path of practice for the cessation of kamma should be known.' Thus it has been said. In reference to what was it said?
"Intention, I tell you, is kamma. Intending, one does kamma by way of body, speech, and intellect.
"And what is the cause by which kamma comes into play?
Contact is the cause by which kamma comes into play.
"And what is the diversity in kamma? There is kamma to be experienced in hell, kamma to be experienced in the realm of common animals, kamma to be experienced in the realm of the hungry shades, kamma to be experienced in the human world, kamma to be experienced in the world of the devas. This is called the diversity in kamma.
"And what is the result of kamma? The result of kamma is of three sorts, I tell you: that which arises right here and now, that
which arises later in this lifetime, and that which arises following that. This is called the result of kamma.
"And what is the cessation of kamma? From the cessation of contact is the cessation of kamma; and just this noble eightfold
path -- right view, right resolve, right speech, right action, right livelihood, right effort, right mindfulness, right concentration -- is
the path of practice leading to the cessation of kamma.
"Now when a disciple of the noble ones discerns kamma in this way, the cause by which kamma comes into play in this way,
the diversity of kamma in this way, the result of kamma in this way, the cessation of kamma in this way, and the path of practice leading to the cessation of kamma in this way, then he discerns this penetrative holy life as the cessation of kamma.

Dependent origination states (paṭiccasamuppāda):

And what is dependent co-arising? From ignorance (2nd noble truth) as a requisite condition come formations (kamma, 2nd noble truth). From formations as a requisite condition comes resultant-consciousness (start here to feeling, all is the 1st noble truth). From consciousness as a requisite condition comes nāmarūpaṃ. From nāmarūpaṃ as a requisite condition come the six sense media. From the six sense media as a requisite condition comes contact. From contact as a requisite condition comes feeling. From feeling as a requisite condition comes craving (2nd noble truth). From craving as a requisite condition comes clinging (2nd noble truth). From clinging as a requisite condition comes becoming-cause (kamma, 2nd noble truth). From becoming-cause as a requisite condition comes birth (start here to feeling, all is the 1st noble truth). From birth as a requisite condition, then aging & death, sorrow, lamentation, pain, distress, & despair come into play. Such is the origination of this entire mass of stress & suffering.

Avijjā and taṇhā are co-arising. See SN 22.81:

"Well then — knowing in what way, seeing in what way, does one without delay put an end to the effluents? There is the case where an uninstructed, run-of-the-mill person — who has no regard for noble ones, is not well-versed or disciplined in their Dhamma; who has no regard for men of integrity, is not well-versed or disciplined in their Dhamma — assumes form to be the self. That assumption is a fabrication. Now what is the cause, what is the origination, what is the birth, what is the coming-into-existence of that fabrication? To an uninstructed, run-of-the-mill person, touched by that which is felt born of contact with ignorance, craving arises. That fabrication is born of that. And that fabrication is inconstant, fabricated, dependently co-arisen. That craving... That feeling... That contact... That ignorance is inconstant, fabricated, dependently co-arisen. It is by knowing & seeing in this way that one without delay puts an end to the effluents.

Paticcasamuppāda is loop.

(past:) having kilesa (especially avijja&taṇha) > do kamma (saṅkhāra/kamma-bhava) > (present:) becoming/born (suffering/uppatti-bhava/jāti) > having kilesa > do kamma > (future:) becoming/born > (infinity loop).

Please see this answer for an in-depth discussion on this topic.

The Ekamsena Sutta and this "Kamma: A Study Guide" article does say that past kamma results in consequences.

However, we understand from the Sivaka Sutta (quoted below), that not everything we feel is due to past kamma. In this answer, you can find that the Buddha himself felt painful sensations due to dysentery but it's not due to his kamma becoming ripe.

[The Buddha:] "There are cases where some feelings arise based on bile. You yourself should know how some feelings arise based on bile. Even the world is agreed on how some feelings arise based on bile. So any brahmans & contemplatives who are of the doctrine & view that whatever an individual feels — pleasure, pain, neither-pleasure-nor-pain — is entirely caused by what was done before — slip past what they themselves know, slip past what is agreed on by the world. Therefore I say that those brahmans & contemplatives are wrong."

"There are cases where some feelings arise based on phlegm... based on internal winds... based on a combination of bodily humors... from the change of the seasons... from uneven care of the body... from harsh treatment... from the result of kamma. You yourself should know how some feelings arise from the result of kamma. Even the world is agreed on how some feelings arise from the result of kamma. So any brahmans & contemplatives who are of the doctrine & view that whatever an individual feels — pleasure, pain, neither pleasure-nor-pain — is entirely caused by what was done before — slip past what they themselves know, slip past what is agreed on by the world. Therefore I say that those brahmans & contemplatives are wrong."

In the next part you ask:

OP: Mostly we have a very detailed description of the 1st noble truth: life is suffering (sickness, death, old age, the whole life disappointments, departures, not getting what person wants, losing something/someone...etc) And a jump to cravings/desires explaining that suffering in the 2nd noble truth. But imho it doesn't explain all the suffering, only some.

Life is not suffering. There is suffering. Perhaps you can say that existence is suffering and that would be more accurate, but a detailed study would show that the Buddha taught that "there is suffering" but did not say that it is life that is the suffering or that it is existence that is the suffering. If it were, then you can simply end that suffering by suicide. This also relates to the question of who is suffering and because of whose actions.

Life is sometimes happy and sometimes sad. Both the happy and sad times do not last forever. It's not always sad.

Craving is not the direct cause of suffering. How craving is the cause of suffering, can be understood through dependent origination. Please see here for details.

Sickness is not mentioned in the 2nd noble truth. The 2nd noble truth mentions three types of craving, namely, craving for sensual pleasure, craving to be & craving not-to-be as causes of suffering.

The 2nd noble truth explains suffering arises in relation to sickness due to craving not-to-be sick.

Of course, there is karma, child is sick because of previous karma.

Previous karma does not create sickness. Sickness is created by virus, bacteria, etc. For example, a person might get sick because of the negligent kamma of not taking a precaution, such as not wearing a condom when having sex. But the sickness is created by bacteria and not by kamma. For example, modern people who have lots of promiscuous sex do not get sick as often as they used to because of modern medicine. Where as in the past, promiscuous sex resulted in people dying from syphilis, gonorrhea, etc. This shows sickness is not caused by kamma.

A child gets sick because of bacteria or because of the negligent kamma of its parents that results in the child contracting a bacterial infection.

Mostly we have a very detailed description of the 1st noble truth: life is suffering (sickness, death, old age, the whole life disappointments, departures, not getting what person wants, losing something/someone...etc)

The 1st noble truth does not say life is suffering. The 1st noble truth summarised all suffering as attachment to the five aggregates. Sickness is include here because it is common for people to suffer about sickness. But it is due attachment to sickness that people suffer, namely, craving not to be sick. Sickness itself is not suffering. The Buddha got sick but did not suffer when the physical body was sick.

In Buddhism, there is physical sickness & mental sickness. The 2nd noble truth is about the arising of mental sickness (dukkha). Please read SN 22.1, which states when the body get sick, to not allow the mind to become sick.

So it is, householder. So it is. The body is afflicted, weak & encumbered. For who, looking after this body, would claim even a moment of true health, except through sheer foolishness? So you should train yourself: 'Even though I may be afflicted in body, my mind will be unafflicted.' That is how you should train yourself. SN 22.1