It is known that there is a model structure on category of dg algebras (non-commutative over arbitrary commutative ring). In particular it is complete and co-complete category. My question is how to construct limits and co-limits. I'm especially interested in co-equalizers. 



Coequalizer in category of dg-algebras
There is a general bit of category theory that was applied to 
ring spectra in EKMM ([83] on my website) and I'll refer
to that for details. Unless I am missing something, the
discussion surely specializes just as well to dg algebras over a 
commutative ring $R$. I'll outline the recipe it gives
for constructing all colimits of dg $R$-algebras as reflexive 
coequalizers in the category $Ch\_R$ of chain complexes over $R$.


The free graded $R$-algebra functor induces a free dg $R$-algebra 
functor on $Ch\_R$. That gives a monad $T$ on $Ch\_R$ whose algebras 
are the dg $R$-algebras. This monad preserves reflexive coequalizers 
by Prop. 7.2, p. 47. Therefore, by Lemma 6.6, p. 46, if $g\colon B\to C$ 
is a reflexive coequalizer of maps $e,f\colon A\to B$ in $Ch\_R$ such that 
$A$ and $B$ are $T$-algebras and $e$ and $f$ are maps of $T$ algebras, then
$C$ has a unique structure of $T$-algebra such that $g$ is a 
map of $T$-algebras, and $g$ is the coequalizer of $e$ and $f$
in the category of $T$-algebras. Now all colimits in the
category of $T$-algebras are constructed from just such 
reflexive coequalizer diagrams in $Ch\_R$, as shown in the
proof that the category of $T$-algebras is cocomplete given in
Prop. 7.4, p. 49. 


