Are wild cats the same species as house cats?
There are many definitions of species - look at the wikipedia page for an overview (this is a large and tricky subject). Some hybridization is not uncommon between "proper" species (often in narrow contact zones), also with fully or partially fertile hybrids. Definitions of species in these cases often comes down to genetic similarity/dissimilarity of populations. Even though hybridization is taking place it usually does so at very low rates, so the gene pools stay relatively distinct. The biological species concept (which is what you are referring to) is also problematic when it comes to e.g. ring species. 

As for wild cats and domestic cats, my feeling is that modern evidence (e.g. molecular studies and phylogenies) places domesticated cats firmly as a subspecies of wild cats (Driscoll et al. 2007, 2009), with the domesticated cat as Felis silvestris catus. In the study by Driscoll et al. (2009), domesticated cats fall in the same mtDNA clade as F. silvestris lybica. However, domesticated cats were traditionally described as a separate species (by Linnaeus), and this has stuck and some still label them as separate species (as does Mammal Species of the World). I can imagine that some who argue for species status now might use a mixture of ecological, phenetic, and biological species definitions.

From Driscoll et al. (2009):


  In contrast, the world's domestic cats carried genotypes that differentiated them from all local wildcats except those from the Near East. Domestic cats show no reduction in genetic diversity compared with the wild subspecies (37), thus giving no indication for a founding genetic bottleneck. Multiple genetic analyses produced concordant results, in each case tracing the maternal origins of cat domestication to at least 5 wildcat lines (A through E, Fig. 2B) originating in the Near East. The domestic cat is referred to as a sixth subspecies, F. silvestris catus, although it is clear that domestic cats derive very recently from F. silvestris lybica (37). 
  
   ... the vast majority of sampled domestic cats fall into the same mtDNA clade, which also includes F. silvestris lybica...


There is also evidence that hybridization can help adaptation by facilitating gene transfer (even though hybrid fertility/viability might be low), so you might want to check out e.g. results from the Heliconius Genome Consortium (2012).