Question 1: The model category C $\mathcal{C}$ should be left proper: , i.e. the pushout of a weak equivalence along a cofibration is again a weak equivalence. (Dually, there is a notion of right proper.) Top is left proper, as is any model category in which every object is cofibrant, such as SSet. There is some information on this notion of properness at the nlab, and I think it's also discussed more thoroughly in Hirschhorn's book Model Categories and their Localizations (and probably many other places).