Sure, or his grandfather taught him how to shoot against his father’s wishes. Whatever story you want.
And though crossbows make more sense, I just personally don’t like them. Missile weapons aside, having some background in sword would be more reasonable than him being insta-proficient just by picking up a sword
I think he would certainly know a thing or two about swords. Being around them your whole life, practicing with wooden sticks like we used to do in our youth. You can’t make a good sword without at least knowing how to balance it for a good swing.
PS, it seems the pope succeeded in making the crossbow seem like a dishonorable weapon 
Henry definitely could have some decent knowledge of technology, quality and craftsmanship (as demonstrated in the livestream preview), depending on how much his father taught him.
For the actual use of the weapon, that is pretty hard to say. He could probably know some basics, if someone taught him that as well, but there’s not much guarantee of his proficiency in combat.
Smiths usually didn’t just make swords every day as their primary product. Most of the regular smithing work actually covered various utility tools, parts and devices for everyday use and unless you were a specialized weaponsmith based somewhere where it would be a good bussiness, you would make weapons like swords only occasionally, as a speciality. Henry’s father may have been a regular village smith, probably spending most of his worktime on stuff like horseshoes and making a sword only occasionally as a personal masterpiece to give to his lord for some extra recognition and money.
The livestream game prototype was primarily a technological demonstrator to show off the mechanics, the player character was even immortal, so it doesn’t tell much about the main char’s skills at that point in the actual intended story. But considering the fact that he was supposed to receive some serius beating and almost get killed while trying to just find and bury his parents (wich is given by the story itself) shows that he was no real fighter yet.
Having some option to choose a few small skill bonuses (like 5-10%) tied-in with some minor background choices for a bit more character customization doesn’t really sound like a bad idea, though. No matter how much they really taught him, Henry is still supposed to have well over 20 years of life behind him already.
You misunderstood my point. Blacksmith can do just horseshoes his whole life. I bet that making horseshoes, agricultural tools or the wheel arches is a bit different from making swords 
And in that case he would know nothing about swords at all. And like others already pointed. Ability to create something doesn’t mean that you can actually use it properly.
I would not trust a blacksmith to make me armor or a sword if he never held or wore one before, just saying.
Craftsmanship does not imply proficiency. I’m sure as a blacksmith, knowing a bit about swordsmanship is important. But knowing how to reposte or parry (efficiently) or pirouet or your foot stance? Not critical to the formation of the blade. That’s why master craftsmen were named so. They weren’t master swordsmen too.
