No, the virus cannot reproduce.

It simply does not have the parts to reproduce. Only instructions on how to produce it. A virus is even worse at reproducing than Ikea chairs:, the chairs bring both the parts and the manual, the virus only comes with the manual. The extremely specific conditions you mention do not allow the virus to reproduce, it allows the host cell to produce the virus.

You might not like the definition of the word, but it is what it is.

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The virus can reproduce, it just requires a very specific environment to do so, including specific molecules that are only produced by other life, such as ribosomes.

Humans require very specific environments to survive, including a long list of chemicals that are only produced by other life, such as vitamins.

The requirements for the virus are a lot more specific, but there is nothing fundamentally different in them.

By these definitions for eat and grow, a fire is alive. I'd consider prions no more alive than fire, while still being as dangerous as fire.

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It does have just genes for sex determination, so it does not need to be big because it carry not as much information as any of the other chromosomes.

You do not need two X chromosomes active, in fact, in women one of the X chromosomes is inactive and condensed in a Barr body.

This is not a universal sentiment in the biology community and some like myself count viruses as oganisms. And I maintain not counting them is very very stupid. They have lineages, code which evolves over time, and self replicate in the right conditions. Just like every other "organism". If we're counting obligate parasites' as organisms, there's no reason to not count viruses. And there are some viruses bigger than some bacteria with thousands of genes.
Prions on the other hand don't really evolve, it's just the same misfolded protein constantly.

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> Also sex-determination doesn't only involve the Y chromosome, but around 60 genes working in concert all over the genome.

But aren't all of those triggered by SRY, which is (usually) located on the Y-chromosome?

The Y chromosome is so much shorter than the X because there are much fewer genes & much less genetic information stored in that chromosome. This is because the Y chromosome can’t have any absolutely essential genes, because roughly half the population doesn’t have a Y chromosome and still needs to fully function & survive. The Y chromosome mostly contains genes related to triggering male development, but the genes that actually control male development are on many different chromosomes (including the X).

Everyone has at least one X chromosome, and in fact the X chromosome does contain some essential genes, which also speaks to its length.

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Isn't this basically the "RNA world" hypothesis? There should definitely be papers on that. Recently there was a new paper on the question how it could actually have been possible for the RNA bases to emerge from inorganic molecules. For a long time the RNA world hypothesis was (is) very popular yet from a chemistry point of view it is quite hard for RNA bases to form from inorganic starting materials (if I understood correctly). Whereas amino acids form comparatively easy from such starting materials. That's why some hypothesized that proteins might have been the origin of life. Yet protein can't replicate themselves. Even prions need existing correctly folded proteins to convert them into their prion state. That's the cool thing about ribozymes. They can self replicate.

I work with viruses, some virologist like 'genetic material replicating and closed in a membrane' which makes me laugh

There is a book by Freeman Dyson called "Origins of life". He talks about the possibility of a self replicating polynucleotide as the supposed origin of life on earth.

But if these would really be the "origin of life" why are they only found in plants?

Short answer: probably.

Long answer: all life, the simplest example of which is the cell, must be able to locally reduce entropy while increasing the entropy of it's environment. So far that we're aware, this includes metabolic processes, which take simple (comparatively) molecules, restructure and recombine them, and poop out less energetic small molecules. To do this requires big molecular machines, or proteins and enzymes. These big molecules are coded like a computer (in this metaphor, it may be helpful to think of the protein/enzyme as executable code, etc.) by DNA, which again for this metaphor may be thought of as binary. The binary must be translated from 1 & 0 to a programming language, RNA in this case. The RNA may then input executable code. This requires tremendous effort, at least 50 separate proteins (again, coded commands) and only makes binary to code language to executable code. That's one pathway. One fundamental requirement. When you ask about energy metabolism like fermentation, or membrane construction, or things like that, the process gets even more whackadoodle. This is to give you an idea of the complex requirements for the basic concept that life locally reverses entropy. Now consider that for each amino acid of a protein/enzyme (the big machines, the executable code) there are 3 base pairs of DNA called a codon. Each protein can have between 100 and 1 million amino acids. That means that, not counting things like RNA, a genome must be massive for a living thing to actually be alive. Indeed, the smallest confirmed genome of a living thing was isolated from an endosymbiotic organism (a critter that lives in the cell of another critter), Nasuia, that has around 190,000 base pairs of DNA. This thing can only synthesize around 10 amino acids using their DNA. So that's probably the minimal genome possible.

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Right? The concept of "life" is surprisingly tricky. But I personally would not consider prions to be alive.

I will say though, prions definitely "eat", when they destroy the normal form of the protein. They "grow" by increasing their population, much like bacteria / viruses. Breathing is not a requirement for life.