In the direction of question 1: if $X$ is compact Hausdorff and $Y$ has the homotopy type of a CW complex, then two maps $X\to Y$ are C-homotopic if and only if they are homotopic. In particular, C-homotopy groups agree with usual homotopy groups for CW complexes. Edit: As Harry pointed out, the definition of homotopy groups uses relative homotopy, so an equivalence of C-homotopy and homotopy does not imply C-homotopy groups agree with homotopy groups.
For $X$, $Y$ arbitrary, let $\mathcal{C}(X,Y)$ be the space of continuous functions from $X$ to $Y$, equipped with the compact-open topology. If $X$ is locally compact Hausdorff, then for every $Z$ there is a bijection $$ \{\text{continuous maps }Z\to\mathcal{C}(X,Y)\}\leftrightarrow\{\text{continuous maps }X\times Z\to Y\}. $$ This implies that for $X$ locally compact Hausdorff, C-homotopy agrees with homotopy for maps $X\to Y$ if and only if every connected component of $\mathcal{C}(X,Y)$ is path-connected.
When $X$ is compact Hausdorff and $Y$ has the homotopy type of a CW complex, it is a theorem of Milnor that $\mathcal{C}(X,Y)$ has the homotopy type of a CW complex. It follows that the components of $\mathcal{C}(X,Y)$ are path-connected (this is true for CW complexes because they are locally path-connected, and a homotopy equivalence induces a bijection both on components and on path-components).
That said, connected components being path-connected is a much weaker condition than having the homotopy type of a CW complex, so probably there are much weaker hypotheses on $X$ and $Y$ under which C-homotopy agrees with homotopy.