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I'm finding information in Google with the keyword (C2H5OH + K) to find the reaction between $\ce{C2H5OH}$ and $\ce{K}$ and found very little information about it. Just 2-3 related results with contradictory pieces of information.

The first one gave

$$ \ce{K + C2H5OH -> KOH + C2H5}, $$

the second one gave

$$ \ce{2C2H5OH + 2K -> 2C2H5OK + H2}, $$

and the third one gave

$$ \ce{C2H5OH + KOH -> C2H5OK + H2O} $$

There are no trusted public sources so I'm very baffled. I think that $\ce{C2H5OH}$ will react with $\ce{K}$ to form $\ce{C2H5OK}$ because it looks like the reaction with $\ce{Na}$. But I'm not sure and just want to find a trusted source. Can anyone confirm the true result for this reaction?

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You can view water as the simplest alcohol, and indeed, water reacts with all of the Group 1 metals to form hydrogen gas and the corresponding metal hydroxide according to the following equation

$\ce{2M + 2HOH -> 2MOH + H2}$

where $\ce{M}$ is the metal.

The same reaction occurs with simple alcohols to produce the corresponding metal alkoxide and hydrogen gas

$\ce{2M + 2ROH -> 2MOR + H2}$

As you increase the hydrocarbon portion of the alcohol, or reduce accessibility to the $\ce{OH}$ portion of the alcohol (e.g. make the alcohol less like water), you slow down the rate of the reaction.

Here is a nice link that compares and discusses the relative reactivities of the Group 1 metals with water.

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When reacted with alkaline metals, alcohols act somewhat like acids, oxidizing the metal and releasing hydrogen gas while the rest of the molecule becomes an anion. In this case, the anion consists of a negatively charged alkoxy group, called an alkoxide or alcoholate ion.

The equation of the reaction is indeed

$$ \ce{2C2H5OH + 2K -> 2C2H5OK + H2} $$

where the product is named potassium ethoxide or potassium ethanolate.

AFAIK, the reaction $\ce{C2H5OH + KOH -> C2H5OK + H2O}$ does happen as well as the reverse reaction: potassium ethoxide hydrolyses into ethanol and potassium hydroxide. So:

$$ \ce{C2H5OH + KOH <-> C2H5OK + H2O} $$

They exist in an ionic equilibrium with hydroxide being the favored ion, because water has a higher $K_a$ and thus a higher tendency than ethanol to remain in ionic form.


Wurtz reaction

About the other reaction $\ce{C2H5OH + K -> C2H5^{.} + KOH}$ which is provided in the question, I'm not sure if this reaction actually occurs. However, if the hydroxyl group is replaced by a halogen atom, that is in the case of a haloalkane/alkyl halide instead of an alcohol, this reaction actually happens, and is known as the Wurtz reaction:

$$ \ce{CH3CH2Cl + K -> CH3CH2^{.} + KCl} $$

the ethyl radicals produced would then dimerize to form butane:

$$ \ce{2CH3CH2^{.} -> CH3CH2CH2CH3} $$

So the overall reaction would be:

$$ \ce{2CH3CH2Cl + 2K -> C4H10 + 2KCl} $$

This reaction can be used to synthesize structurally symmetrical alkanes, linear or branched(e.g. using 2-chloropropane).

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