Color coding to get an FPT algoirthm for $k$ disjoint triangles
No, this is not correct.

As a counterexample, suppose we have a graph consisting of a central triangle, together with 3 outer triangles, such that each outer triangle is joined to the central triangle by one shared vertex (i.e., each vertex of the central triangle is identified with one vertex of one of the outer triangles).

Clearly, a solution for $k=3$ would take the three outer triangles and not take the inner triangle.

Assume the random coloring assigns a distinct color to every vertex (otherwise there is no colorful solution).

If your greedy algorithm considers the central triangle first it will always take it, but this is incorrect.