[ I now realize that the OP is asking about functors in the image of the forgetful functor from simplicial spaces to semisimplicial spaces, though I focus my answer mainly on a general semisimplicial space. I'll keep the answer up anyway in case it's helpful.
To clarify, by a simplicial blah, I'm as usual talking about a functor $\Delta^{op} \to Blahs$. A semisimplicial blah is given by a functor $\Delta_0^{op} \to Blahs$, where $\Delta_0$ is the non-full subcategory where we discard the non-identity surjections.]
The degeneracies are critical when considering products--it is in fact false that the geometric realization commutes with products (up to homotopy equivalence) without the degenerate simplices. A simple example is given by taking the product of $S^1$ with itself---if you take the one-vertex, one-edge semisimplicial structure, you'll never recover the torus via geometric realization. (An obvious problem arises when now your higher simplices $X_n$ can equal the empty set.) Note this is an example in semisimplicial sets, not even spaces. In short, the functor $||\bullet||: semiSSpace \to Spaces$ is not well-behaved with respect to products.
However, these "incomplete" simpicial sets/spaces arise really naturally. For instance, a lot of categories may not have a natural unit/identity, so there are no degeneracy maps in the nerve of the category---these are modeled most easily by "semi"simplicial sets/spaces. But so long as you're not taking products, the theory of such things (I think) work out just fine. You can still talk about the classifying space of a non-unital category by taking geometric realization of the corresponding semisimplicial set (i.e., its nerve).
As for your second question, I'm fairly certain that what you say is correct--if you have two semisimplicial spaces $Y_\bullet, X_\bullet$ with a natural transformation that induces level-wise equivalences, then the geometric realizations will be (weakly) homotopy equivalent. I think you can see this by taking a filtration by simplicial index and seeing that the associated gradeds are equivalent.
Given a semisimplicial space, there are some ways to get an actual simplicial space (this is akin to formally adding units in a category, or to an algebra) but I'm not sure how well-behaved this functor is.
However, the composite functor $$ SSpace \to semiSSpace \to Spaces $$ (forget, and then realize) will satisfy all the properties you asked for, mainly because the fat geometric realization doesn't differ in homotopy type from the usual geometric realization.
As always if anybody has questions or more enlightening comments, please share.
Hiro