[ 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.]
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.
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.
As always if anybody has questions or more enlightening comments, please share.
Hiro