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Imagine a large and massive dust cloud made of solid phase micrometeoroids, asteroids and larger planetoids, all material in it is in solid phase, and contains no hydrogen or other volatiles in gas form - think about the chemical composition of inner solar system asteroids.

Can this debris cloud collapse and form a star that fuses the lighter than iron elements? What properties such object would have? Would it be very hot? Very cold?

Nowadays it's unlikely, but I can imagine situations like this near the end of the stelliferous era when most of the hydrogen is used up and the universe is filled with white dwarves, neutron stars, and black holes, and lots of dust from all the previous explosions of the supernovas and planetary nebulas of the last hydrogen stars.

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    $\begingroup$ There's some relevant material in the answers to this question: astronomy.stackexchange.com/q/33963/16685 $\endgroup$
    – PM 2Ring
    Commented Nov 29, 2019 at 22:49
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    $\begingroup$ Note that silicon burning normally involves helium, (but at those temperatures, helium will be produced via photodisintegration) and is a rather short-term process in normal stars, so your "rock star" won't do fusion for very long. $\endgroup$
    – PM 2Ring
    Commented Nov 29, 2019 at 22:56
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    $\begingroup$ In principle the solid dust could stick and, depending on composition and overall mass, ignite. If this would be a star sustaining process or a kind of explosion is out of my basic knowledge. It might depend on what solid element was in the dust cloud, too. $\endgroup$
    – Alchimista
    Commented Nov 30, 2019 at 6:55
  • $\begingroup$ It would be somewhat like the core of a red giant. Look for models of red giants. $\endgroup$
    – Incnis Mrsi
    Commented Dec 22, 2020 at 19:57

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