Not to try to bring back a dead and disliked comment, but technically a male can have children. I hate to shit on the heavily believed mindset that a man cannot have kids, but it is possible, albeit with heavy scientific intervention which is still being researched and developed in modern day society. Most notably the Japanese scientists are trying to create Female Sperm, yes you heard me, Female sperm. I believe it's created using various cells in the body, most likely stem cells, but I'd have to do some research on that to be 100% certain. Such technology has already successfully been implemented on Chickens, so the Human side of such technological advancements isn't far off.

But I digress; in regards to the lore it's speculated that Seathe created the lower life-forms known as the Drakes in his attempt to gain immortality as a Dragon and possibly to gain some form of kin, but again that's speculation. Regardless, if he could create an entire species it's not hard to believe that he could create a child from a mans cells. Oh and I forgot to mention, it's a game, nothing is impossible!

I doubt that the Serpant Soldiers are incomplete dragons, but more-so either a scientific creation created by Seathe, or a species that existed before the lore began (which I personally find more likely). The Drakes, as I stated above, are speculated to have been created by Seathe in an attempt to discover immortality and possibly to create fellow kin after his sold the Dragon's Weakness to Gwyn. If you look at the Drake Sword's Flavor Text you can see that it states:

"This Sword, one of the rare dragon weapons, is formed by a drake's tail. Drakes are seen as undeveloped imitators of the dragons, but in they are likely their distant kin."

What you can take away from that is that they're either a distant kin of the Ancient Dragons that existed before the Age of Fire began, or they were created by Seathe in one of his many experiments.

tl;dr: Men can have children with the help of scientific intervention, so it is possible that Seathe created such magic to do that for him. Drakes are either a distant kin of the Ancient Dragons or they were created by Seathe.