How do we know that the dark matter is cold or non-relativistic? According to the $\Lambda$CDM parametrization of the Standard Model of Big Bang cosmology, the universe contains a cosmological constant $\Lambda$ associated with $73\%$ dark energy, $23\%$cold dark matter (CDM) and $5\%$ ordinary matter. 
I have also read from Kolb and Turner's book that the Cold dark matter leads to a "bottom-up" formation of structure in the universe while hot dark matter would result in a "top-down" formation scenario; since the late 1990s, the latter has been ruled out by observations of high-redshift galaxies such as the Hubble Ultra-Deep Field.
$\bullet$ How do we know that the dark matter is cold or nonrelativistic?
$\bullet$ What is the percentage of hot dark matter in $\Lambda$CDM parameterization? Is the possibility of dark matter totally ruled out?

Addendum Existence of CDM does not rule out existence of HDM. In fact, it is assumed that the dominant fraction of the dark matter is CDM type. But observations only tell us that $23\%$ of the total mass-energy of the observable Universe dark matter. But this information does not tell how much of it is HDM and how much is CDM? The belief is that the the dominant component of the dark matter is CDM but is there an estimate? Is there a physical way to understand/expect why the CDM should be the dominant contribution?
Please note There is a typo. I wanted to ask "Is the possibility of hot dark matter totally ruled out?"
 A: EDIT: Question's changed a bit since I answered. I'll leave my old answer below, but the question of what can be "known" about the early universe is perhaps best left to philosophers. The best we can get in science is what theories are most consistent with the observations we have.  Given current information, observations support theories in which dark matter remains coupled to ordinary matter for a relatively long time (Cold Dark Matter) somewhat better than they do theories in which they decouple earlier (Hot Dark Matter).  But until footling little issues like Inflation are thoroughly explained, I'd be rather cautious about claiming we "know" one way or the other. 
Original Answer:
Dark matter is often assumed to be cold because if it were hot it would radiate, and hence be visible ('light' matter, perhaps?). Not all theories demand it be cold, some versions of WIMPs would not radiate simply because they do not interact with the electromagnetic force. 
Non-relativistic (I assume you mean velocity << c?) comes from the apparent distribution of dark matter inside galaxies: For the matter to be where it needs to be to give the mass distribution inferred from the orbits of the visible matter, it must have a non-relativistic orbital velocity. Assuming it's orbiting of course, but if its not then we need a very exotic explanation of how it got there.
I don't know how you could ever totally rule out the possibility of dark matter - even if you could prove a theory that does not invoke dark matter to explain the various missing mass observations (MOND for example), you couldn't prove that there wasn't still a small amount of dark matter hidden in some corner of the universe. 
A: To answer shortly to your question lets just say that when you look at the high red shift universe you're in a sense looking into the past because you're looking at parts of the universe "closer" to the Big Bang. Knowing the universe at that redshift and knowing our own (redshift 0) gives us 2 points in "time" which show that the structure in the universe forms in a bottom up way. So dark matter must be cold. Nevertheless there are several simulations that show that such a structure formation scenario is possible with warm dark matter instead of cold.
