Disclaimer: I'm in fact not sure where you stand on rebirth, whether you deny it or just dismiss as unimportant.
I consider 'birth' ('jati') to refer to the production of the view ('ditthi') of 'a being' ('satta'), per SN 12.2, SN 23.2, SN 5.10, SN 22.81, etc. Therefore, I do not deny 're-birth', which is the re-production of this self-view or self-identity (MN 44) of 'a being', both internally & externally. My personal research on 're-birth' can be read at this link.
1. If there's no rebirth, what happens after death of a non-arahant? Does a new life ensue dependent on the one which has just ended?
'Death' ('marana'), similar to 'birth' ('jati'), is a self-view about the death of 'a being', per SN 12.2, SN 23.2, SN 5.10, etc.
The sutta says arahants are not only not subject to 'birth' but they are also not subject to death (per MN 140; SN 22.85). The Dhammapada states arahants do not die but non-arahants are dead already.
Given the questioner is also a student of Xianism, they should understand 'birth' (per Nicodemus) & 'death' do not always have a physical meaning.
Therefore, the minds of non-arahants are subject to birth & death of self-views & self-identities.
As for a new life ('jiva') after the termination of life, this idea or speculation arises in minds that cling to & crave for life. It is a natural instinct that people do not want to die and wish for new life or eternal life, which is why religion is very popular.
2. Rebirth is tightly connected with kamma.
Absolutely. MN 117 states kamma & results ('rebirth') is a mundane teaching siding with merit & attachment. What this means is the results of kamma is always taken personally, such as "I did good", "I did bad", "I am happy", "I am unhappy". Kamma has self-view, unless it is noble kamma that ends kamma (per AN 6.63).
a) Do you observe ripening of all results of wholesome and unwholesome actions in the current life?
Noble practitioners have ended kamma. However, result of kamma can only be observed in this life. SN 12.25; AN 6.63, etc, state there can be no kamma without contact.
b) If you don't, what's in your opinion the fate of those kammas which haven't produced result in the current life?
All kamma produces a result in the present & also future. For example, if my mind gets angry or annoyed, my body & mind become stressed. This stress is a result of kamma. The most small act of clinging produces a result of dukkha, which is why MN 140 & elsewhere states conceiving 'self' is a dart, a tumour, a cancer. In the 1st noble truth, the Buddha summarised 100% of dukkha as 'upadana' ('attachment'). But in MN 26, Buddha said this generation that delights in attachment cannot see attachment is dukkha. That is why this generation believes attachment to a wife, husband, girlfriend, boyfriend, child, etc, is happiness.
3. Current existence is traditionally thought of as being a manifestation of kamma amassed in previous existences.
No. I do not believe a Buddha can teach this. It is only a view of those who reject the Dhamma is visible here & now.
The sutta (SN 35.145) about "old kamma" refers to the sense organs (not the aggregates) and says they "ought to be viewed as old kamma", which means the noble practitioner should not let the mind drift back to the past, thinking about old kamma but, instead, focus on the sense organs.
If there's no rebirth, current life hasn't been preceded by any other existence which would contribute its kamma to it.
This question is irrational because we know life comes from the sperm & ovum of our parents & we know the kammic or otherwise random events that have lead to our fortune & misfortune in life. Kamma is intention. A new born child is only born with very primal instincts. Please refer to MN 64 about a new born child.
How do you then explain differences between people and their experiences, which in MN 135 in particular are explained through kamma?
MN 12, SN 14.15, etc, say the difference among people occur due to the 'elements' ('dhatu'), just as the modern science of nature (genetics) vs nurture. AN 3.61 states present life experience is not due to past kamma.
There is no evidence the Buddha spoke MN 135, considering its views are only found in two suttas. Also, MN 135 refers to worldly conditions rather than to happiness & suffering, which the other kamma teachings are about. Also, the worldly conditions in MN 135 are found in many other suttas as states of mind, such as spiritual beauty, spiritual wealth, etc. Also, if MN 135 referred to physical conditions, it is obviously false, since the most wealthy people in the world have acquired their wealthy via very evil means. It is ridiculous to say the Rockefellers, Rothschilds, Bush Family, George Soros, etc, are wealthy due to past good kamma. Rational people understand the wealthy families of the world often gained their wealth via colonialism, exploitation, murder & evil. Many of the most wealthy people & families in the world engage in wrong livelihood.
Regardless, MN 135 is not the Dhamma. It is a sutta spoken to a non-Buddhist Brahman student and anyone who takes it seriously cannot, imo, be considered a noble Buddhist.
4. If a person doesn't attain Nibbana in the current life, isn't there any hope for them?
The Dhammapada states the heedless are as if dead already. The Buddha remained silent when asked would he save all sentient beings. The last chapter of the SN states all beings who do not realise the 4NTs will take rebirth in hell (dukkha). The Dhammapada compares the people of the world to a rubbish heap.
If there was a next life, those who do not reach stream-entry in this life would not get closer to Nibbana in a future life because no amount of morality alone can result in stream-entry.
I have not read any suttas where the Buddha taught puthujjana laypeople they might reach Nibbana in future lives. The Buddha did not teach about Nibbana to puthujjana laypeople. In his loving-kindness, the Buddha generally taught laypeople if they do good kamma they will have a 'heavenly' ('happy') 'rebirth'. He did not teach them after eons in heaven they will be reborn in hell.
Heedfulness is the path to the Deathless. Heedlessness is the path to death. The heedful die not. The heedless are as if dead already.
Upon a heap of rubbish in the road-side ditch blooms a lotus, fragrant and pleasing.
Even so, on the rubbish heap of blinded mortals the disciple of the Supremely Enlightened One shines resplendent in wisdom.
Blind is the world; here only a few possess insight. Only a few, like birds escaping from the net, go to realms of bliss.
To conclude, I have not read in the Pali suttas the role of Buddhism is to save all sentient beings, let alone seek to convert or attract unsuitable people to Buddhism. Originally, the scriptures say the Buddha determined to teach only those with little dust in their eyes. Buddhism is not the same as Christianity, which condemns non-believers.