For the second question, it sounds like you have the right idea already.
For the first, one way to view the situation is as follows.
In the case where $T$ commutes with colimits, so that the arrow you label $\tilde{T}$ is invertible, it is straightforward to see that the colimit of $D$ is given by $(T\colim(X), Y, f)$ where $f$ is the arrow identified with $\colim(\kappa)$ up to the isomorphism $\tilde{T}$.

Then in the general case, one can think of this pushout as a correction for $\tilde{T}$ not being invertible.
In other words, $f$ is the "closest approximation" to $\colim(\kappa)$ in some precise sense, being by definition the co-base change of $\colim(\kappa)$ along $\tilde{T}$.
(Co-base change is just another word for the arrow obtained by pushing out.)
Of course the precise sense I refer to is just the universal property of pushouts.

Just for completeness, let me sketch the case where $T$ commutes with colimits.  One way (maybe not the easiest one) is to use the identification of the comma category $T\downarrow C$ with the fibred product

  $$\require{AMScd}
\begin{CD}
T\downarrow C @>>> C^{[1]} \\
@VVV @VV{(s,t)}V \\
C\times C @>>{T \times id}> C\times C
\end{CD}$$
where $C^{[1]}$ is the category of arrows in $C$ and $(s,t)$ are the source and target maps.
This follows from the description of diagrams in $T\downarrow C$ given in the statement of your first question, for example.
Hence it is sufficient to understand colimits in categories of the form $A\times_C B$.
There are various ways to do this, when the functors $A \to C$ and $B \to C$ commute with colimits (which they do in our situation when $T$ does).
For example, consider the global colimit functor
  $$ \colim : \underline{Hom}(I, A\times_C B) \to A \times_C B $$
where $\underline{Hom}$ is the internal hom in Cat.
By the assumption one sees that this is identified with the functor on
  $$ \underline{Hom}(I, A) \times_{\underline{Hom}(I, C)} \underline{Hom}(I, B) $$
induced by the global colimit functors in $A$, $B$ and $C$.
The original claim then follows by a direct computation.