It is well known that in the category of all topological spaces, quotient maps aren't preserved by products (this follows from the simpler fact that $X\times (-):Top\to Top$ doesn't preserve quotients). The usual solution, if one is needed, is to change to the category of k-spaces and k-continuous maps. There are other categories where products and quotients 'get along' (e.g. $Set$, $Ab$).
Question 1: What is a large class of categories where quotient maps are preserved by products? Topoi? (Semi)abelian categories? Categories of algebras for a monad on a given category with this property?
Now one may be only interested in a certain class of quotient maps (like surjective submersions in $Diff$, the category of finite dimensional smooth manifolds). Say, regular epimorphisms, or maps admitting local sections (assuming we're in a site), or perhaps something like surjective topological submersions, where there are sections through every point in the domain. So in this case it is not a matter of putting restrictions on $X$ such that $X\times(-)$ preserves quotients, or changing the category, but narrowing the scope of the quotient maps one wants to preserve.
Question 2: Is there are large class of quotient maps (in $Top$, or in a general category - with finite products and enough colimits) that are preserved by products?