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.
There is a general bit of category theory that was applied to ring spectra in EKMM ( 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.