That's right - thanks Eric! So the second part of the question is obsolete.. But I have no idea about the first part.

Good remark, and you're right. $K_2$ is a HC-graph, and any graph that has a perfect matching can be partitioned into copies of $K_2$. My mistake was "more than 2" instead of "more than one" in the formulation of the statement. Just corrected this.

Your answer is very thorough, clear, and useful, so thank you very much!

If you replace the colouring number with the chromatic number, the question is equivalent to the Erdos-Faber-Lovasz conjecture (en.wikipedia.org/wiki/…) -- see arxiv.org/abs/math/0305073

That's right -- sorry for the omissions! Will correct this.

Very interesting, @bof, I suggest you put this as a new question on MO