Why does this construction give a (homotopy-invariant) suspension (resp. homotopy cofiber) in an arbitrary pointed model category? In their text Foundations of Stable Homotopy Theory, Barnes and Roitzheim define the suspension of a cofibrant object X of a pointed model category to be the pushout of the diagram $*\leftarrow X\coprod X\to Cyl(X)$, where the second map is the structure map of the cylinder object. By contrast, there is a more manifestly homotopy-invariant definition of suspension given in e.g. Dwyer and Spalinski, which is the homotopy pushout of the diagram $*\leftarrow X\to *$. It is not clear to me why these definitions agree; if we don't assume properness, I don't even see why the first is homotopy-invariant! (If we assume the model category is proper, then the first diagram's pushout is equal to its homotopy pushout.) There is a similar issue with the cofiber, which they define for a cofibration of cofibrant objects $f:A\to X$ as the pushout of $*\leftarrow A\to X$: again, it is not clear why this is homotopy-invariant (with respect to maps between such $f$ in the comma category) unless the model category is proper. Can we drop the properness assumption and still get homotopy colimits or at least homotopy invariance? Even if so, why are the definitions of suspension equivalent?
 A: 
if we don't assume properness, I don't even see why the first is homotopy-invariant!

The pushout of a diagram A←B→C in which all objects are cofibrant and one of the maps is a cofibration
is always its homotopy pushout in any model category,
see Proposition A.2.4.4 in Lurie's Higher Topos Theory.
This is the case for both of your examples, since the initial object is cofibrant.
A: An argument showing that the two models of suspension are equivalent will probably be based on something like the following:
Assertion:  Suppose we are given a commutative diagram of the form
$\require{AMScd}$
\begin{CD}
\ast @<<< C @= C \\
@VVV @VVV @VV V \\
Y @<<< A @>g>> X \\
@| @VVV @VVV\\
Y @<<< A/C @>>h > X/C 
\end{CD}
in which the vertical directions form cofibration sequences (when I write $A/C$, I mean $A \amalg_C \ast$, where $\ast$ is the zero object), and the maps $g$ and $h$ are cofibrations.
Then   the map of pushouts
$$
Y \cup_A X \to Y \cup_{A/C} X/C
$$
is a weak equivalence, or better still, it is an isomorphism.
It seems to me that this is true by the assumption of properness, since we have a cofibration sequence  given by the pushouts
$$
\ast\cup_C C \to Y \cup_A X \to Y \cup_{A/C} X/C
$$
in which the first term is isomorphic to $\ast$.
Let's call the first suspension $SX$ and the second one $\Sigma X$.
Given the assertion, we can show that the two models for suspension are weakly equivalent as follows:
Apply the assertion to the diagram
\begin{CD}
\ast @<<< \ast\amalg X @= X \\
@VVV @VVV @VVV \\
\ast  @<<< X\amalg X @>g >> \text{Cyl}(X) \\
@| @VVV @VVV\\
 \ast @<<< X  @>>h > CX
\end{CD}
(where $CX = \text{Cyl}(X)/X$)
to get that the map
$$
SX\to \Sigma X
$$
is a weak equivalence.
A: If you're looking to learn more about homotopy colimits, I strongly recommend:

*

*Dugger's Primer on Homotopy colimits

*Shulman's Homotopy limits and colimits and enriched homotopy theory

*Rehmeyer's 1997 master's thesis (under Mike Hopkins), "Homotopy Colimits"

*Homotopy Limit Functors on Model Categories and Homotopical Categories by Dwyer, Hirschhorn, Kan, Smith

*Riehl's book Categorical Homotopy Theory

I note that the first four predate Lurie's books, and the fifth works out many examples. The fact that the pushout and homotopy pushout agree for a span diagram when all objects are cofibrant and one leg is a cofibration (even without left properness) is 13.10 in Dugger's manuscript. A detailed treatment of the cofiber is in Rehmeyer's thesis. Shulman handles your other question, about why these two ways of computing the homotopy colimit agree (e.g., Section 5, drawing on Dwyer, Hirschhorn, Kan, Smith).
