No single episode deals with the difficulties or rationality of cross-breeding among the collective humanoid species of the Alpha Quadrant.
The capability of advanced genetic engineering does exist (Dr. Julian Bashir being an example of genetic physical and mental enhancement) but is considered illegal in the Federation. (DS9 episode: "Doctor Bashir, I Presume?").
The ability to customize genetic development between species is obviously possible and has been so since the early 22nd century (hence Mr. Spock, a Human/Vulcan hybrid survives to adulthood) but perhaps is difficult or limited to those with the proper clearances or access to the level of required technology.
While there does not seem to be a proscription to the practice of hybridization between species, we are unclear why there has not been more energy (or air-time) discussing the process and rationale behind why cross-species hybridization is considered acceptable while single species genetic enhancement is less than desirable.
Yes, I do remember the Eugenics Wars, and do not necessarily believe that is the only reason for holding back the technology. Perhaps the simplest reason is also the best. Cross-species breeding could challenge the very nature of natural species development and might be considered undesirable by more extreme members of any species.
Known genetic manipulations include Cardassian/Bajoran, Human/Vulcan, Human/Betazoid, Human/Klingon and despite the preponderance of these hybrids, none of them would be possible without medical/technical intervention since all of these species have significant differences biologically to warrant such intervention. In some cases, significant genetic modifications would be required.