ct.category theory - Model Structure/Homotopy Pushouts in topological monoids?

Clark Barwick's answer is excellent and you should accept it. This is more of an addendum. The category Top is cofibrantly generated, so $mathcal{C} =$ Mon(Top) is also cofibrantly generated. The key paper is by Schwede and Shipley, and gives conditions on a model category $mathcal{M}$ such that Mon$(mathcal{M})$ is a model category. In the special case of $mathcal{M}$ cofibrantly generated it explains how to get your hands on the cofibrations of Mon$(mathcal{M})$. See Theorem 4.1 on page 8. Of course, now that you have your hands on the fibrations, trivial fibrations, cofibrations, and trivial cofibrations question (2) is also answered. A nice reference for relating the cylinder object to the functorial factorizations is Hovey page 9

Furthermore, every element in Top is fibrant, so the paper above gives you even stronger results, which may help you with your computations. See remark 4.5 on page 10.

The authors also wrote a second paper giving further results. It's here.