The adverb “ευ” as a prefix is inherent as a synthetic in many words. Specifically, eu means "good", as we find it in the words like eulogy, euphemism, euphoria and euthanasia.

The word demon has a long history and today it is used as an oxymoron in relation to its meaning in antiquity which contained semantic piety. In Homer "demons" were originally called the gods, while in Hesiod "guardians of mortals". Thus, the first meaning is "god, goddess". It follows the meaning "fate, destiny, adherent, luck". Therefore, a demon is one who "shares", gives fate to every human being or otherwise who defines it. As is well known, the gods were numerous, but also anthropomorphic. Despite their coexistence, however, the demon declared, above all, an impersonal and indefinable impersonal force, an intangible form. Plato distinguished them from the "visible gods" according to the previous poets, as children of gods in Timaeus. Diotima in Plato's Symposium, speaking of Love, refers to the "genus of demons" as the "intermediate between the mortal and the immortal", that is, in the middle between the two and fills the gap, so that the universe is an indivisible whole ". Through the mediation of demons, the communication between gods and men takes place. Plutarch refers to Socrates the Demon as a kind of higher life guide. Plato, in Socrates' Apology, identifies the demon with a voice within him, "preventing him from doing something, but never urging him". The demon contributes to his faster completion or deification, as long as the mortal perceives the "percussions" and "hears his voice". The demon is commanded to bring back to heaven the fallen man on earth. Plato the Demon is characterized by the immortal-divine part of the soul located in the upper part of the human body and aims through wisdom and truth to lead to blissful life and immortality.

In summary, Eudemonia means that a man has plenty of demons (spirits) that bring him closer to the divine, and while happiness can be a part of this, most remarkable works of our lives will not always bring us joy and satisfaction, but they are well worth pursuing. Actually, Aristotle does not say we should seek eudemonia, but takes it for granted that we already do.

Well while I agree that there are always many things that we don't know, as a neuroscientist I have to disagree that the mind is "merely biologically dependent" and "it does not logically follow".
And to answer the OP question, there is a way to prove that the mind & consciousness exists beyond death. The experiment should show that after the biological material dies, soul & consciousness can be somehow ( I cannot think of a way) transferred (and not copied!) to a new body. I am sure in the future computers will be able to model our cognitive capacities very well but that is merely creating a copy of what we observe of the mind and not proving that the mind exists independent of the biological material.

To even ponder a question like that we need to agree on the definition of words that we use. Like soul, existence and mind.

Per wikipedia:
Soul or psyche (Ancient Greek: ψυχή psykhḗ, of ψύχειν psýkhein, "to breathe", cf. Latin 'anima') comprises the mental abilities of a living being: reason, character, feeling, consciousness, qualia, memory, perception, thinking, etc.
The mind is the set of faculties including cognitive aspects such as consciousness, imagination, perception, thinking, intelligence, judgement, language and memory, as well as noncognitive aspects such as emotion and instinct.
Existence is the ability of an entity to interact with physical or mental reality

There is plenty of scientific evidence that all of the mental abilities described above as a soul & mind have a biological basis and for all practical purposes it is accepted as a fact (at least within the vast majority of the neuroscientific community). And there is plenty of evidence that the soul & the mind seize to exist once someone dies. So the original question is already answered, there is nothing of the sort after death. If then someone argues that existence of a soul includes the memory of him within other people (e.g. my actions are influenced by what my parents would do) then the soul does exist at least partly beyond death, but never the mind or the consciousness.

Reality exists independent of us. We have no way to perceive the majority of things around us. We cannot perceive the UV spectrum as bees can or the majority of the electromagnetic spectrum that is such as Wifi frequencies, radio frequencies and so on. Our visual acuity is not good enough to see anything in the microscopic level. Our sense of smell is pitiful compared to the one that a dog or a mouse has. We hear a tiny fraction of the sound waves. If we sit across a person, we have no idea about the myriad of processes that exist in their head.
We just don't have the sensors to sample everything that exists in the world around us. We "feel" that what we sample is all there is, but there is so much more.

The nervous system does not allow us to perceive reality. It tries to reconstruct the part of reality that is important for its survival. A tiny fraction that is.
We do not have senses that directly measure a specific physical dimension. Even for the 3-dimensional space around us, our brain tries to reconstruct it based on input from the senses. Some senses give more information about the dimensions than others. For example taste/olfaction have no information about 3D space, sound a bit more (for humans because for bats this is the best), vision even more and the somatosensory system is the most accurate when compared to others but is limited in how much can provide at each second.
So coming to the original question, how is time perceived? Or better put, how is time reconstructed? Let's start from the present. As with the spatial dimensions there is no sensory system that directly estimates time. But time information is part of some sensory stimuli like sounds (it after all the changes of sound pressure over time), or vision (movement is change in position over time) and the brain is capable of extracting this information to infer (& perceive) time. There are also plenty of internal rhythms that can be used to estimate time such as breathing, heart rate, attentional shifts, thought progression and lastly specific activation patterns within some brain areas.
Time perception of the past is mostly based on episodic memory and a sequence of events. Also part of the "qualia" of time is the fact that past memories are faded. For example if you can't place something in some temporal time frame with respect to some other events, it should bare no time "qualia".
Finally, perception of the future is mostly the inference of time that we can have based on our models of our every day life. Perceiving "Tomorrow" has to do with things like "Tonight I will go to sleep", "Tomorrow morning I will wake up in my bed", "After that I have drive to work", "For lunch I will cook spaghetti" etc. So based on our previous episodic memories we can have a fairly decent prediction of the future (not too far of course).
There hasn't been found a specific area that processes time information and generates the perceptual qualities of time, but there are probably plenty of areas involved in this such as frontal cortex, parietal, basal ganglia, cerebellum and hippocampus (perception generally involves multiple areas anyways). So there are (most likely) not distinct neurons for the perception of past, present and future.