Asgard is bound to have a messed up foster care system. And to be honest, Loki practically went out of his way to have his kids as weirdly as possible so everyone else must be too uncomfortable to be supportive.
Norse mythology is weird. I'm not sure about the other two but the horse happened because he transformed into a mare to distract a stallion from something so the other gods could win a bet or something like that. Unfortunately for Loki the stallion was faster then he anticipated and caught him, then did what stallions do to mares. So Loki became pregnant and gave birth to an eight legged horse named Sleipnir which in other stories is Odins mount.
Well I guess all those questions can be answered with MAGIC. His two sons Vali and Narfe are 'normal', with Sigyn as mom, but the rest are indeed a mixed lot....and Loki rounding off the brood with giving birth to the last one, Sleipnir