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.