In the first movie, Connor does in fact defeat the Kurgan with heart, soul and steel, as Ramirez says: He's saved by his lover Brenda (heart), demonstrates courage, skill and care for others (soul), and bests the Kurgan in a swordfight (steel).
The only scar sported by an Immortal is the one on the Kurgan's neck, after his head his nearly severed by Ramirez. Since beheading is the only way to kill an immortal, it makes sense that their necks don't heal as completely as the rest of their body.
The Kurgan's loutish behavior in the church in the film. While his savage and thuggish personality is an obvious reason for this, there is another reason lurking in the background. Unlike Connor, who was born into a Roman Catholic society in the early 16th century, the Kurgan was born more than a thousand years before the coming of Christianity. As such, he would consider Christianity as just another glorified cult created by mortals fearful of death. Nonetheless, though he doesn't take Christianity seriously as such, even he still recognizes churches as holy ground and won't fight in them.
In the season 1 finale, "The Hunters," Horton attacks Duncan with a taser in an attempt to immobilize him. But Duncan managed to fight through the voltage and pull the prongs out. The first time I saw this episode, I thought there would be no way he could do that, regardless of how tough he was, because tasers inhibit motor functions. He wouldn't have been able to move. A few years later, I read about the history of the development of taser technology. Around the time this episode was released (1993, I think) the high-powered tasers that inhibited motor function didn't exist yet (those didn't come out until 1998). It was actually very common for people on PCP, or with just a high tolerance for pain, to be able to fight off the older tasers because they were only designed to shock the attacker into submission... which is why the ones we have now use an electrical charge that's the same as the electrical signal the brain sends down to the body, shorting out all muscle control. Basically, if Horton had used one of the modern tasers, the show wouldn't have lasted more than one season.
In the first episode Connor tells Duncan someone should keep an eye on Richie when he witnesses a duel(Duncan agrees and it's how he ends up taking him in). When watching the show the first time it comes of as because he knows about immortals, but the meaning changes when watching it again after the second season episode where Richie becomes immortal. Duncan and Connor both knew Richie was pre-immortal and that's why Duncan took him in, so he could there to teach him if his immortality was awakened.
Duncan is just four hundred years old, and yet he is consistently able to defeat immortals that are older than him by a factor of ten or more. Until we see in a flashback about how Duncan got his first Quickening, an older immortal sacrificed himself to give his power to Duncan when he was still young and inexperienced. We don't how old and powerful that Immortal was. For all we know he could have been as almost as old as Methos. It could definitely explain the edge that Duncan displays in future battles.
Not everyone who comes across an abandoned baby is going to the right thing. In some older cultures, especially, the response might well be to either simply leave or outright kill the baby. So, a newborn baby is violently killed via some method other than decapitation. They regenerate but never physically grow older. Maybe, it's possible that they could eventually grow mentally and might even learn how to speak (babies as young as six months have been known to speak their first words), but could they ever physically be able to sit up, walk, do other things for themselves? Could their body adjust to eating and drinking other things besides milk/formula, or would that be the only thing that they could physically handle without getting sick/suffering pain? And even if they could eventually talk, move, enjoy a diet beyond milk/formula- all that might be worse. They're physically an infant. Forever. It's hinted the immortal in the ten-year-old body was sexually interested in Amanda. No mentally healthy adult would ever be interested in him in such a way, and though sometimes prepubescent kids do engage in mild forms of sexual experimentation with other kids around their age, the idea of a person in a ten-year-old body who is mentally and time-wise much older engaging in such things with an actual prepubescent child is squicky as hell. Now, imagine a person in an infant body developing a sexuality as they mentally grow and having absolutely no way to ever have a safe, healthy sexual relationship with another person. Really, in such a case, a person might argue in all sincerity that the kindest thing to do would be to chop off a baby's head, and- just ugh.
This is actually discussed in a story in the book An Evening At Joes where Methos and company come across a baby who became Immortal after they raided a village. One of them does kill the baby.
Also considering the appalling death rates for children and infants even as recently as the early-20th century, and that there seems to be some indication that an Immortal doesn't need to go through a violent first death... Not only is it conceivable to have scores of Immortal infants and children out there, but the fact that we rarely hear or see of them means that there's very clearly Immortals out there who would have no qualms about taking a child's head (as noted in the "Evening At Joe's" example).