As Flutteryshy mentions in the comments, you probably just missed the enchantment the first time. Enchantments are definitely added when you craft an item, where appropriate. Armor does not get enchantments; weapons and accessories do. This means that it's not a complete waste to find the same things in chests in the caves/sky islands/dungeons over and over because you can find superior enchantments and save gold (see below).
The game will not auto-replace an equipped item. If you find a new copper Broadsword, it will simply go into available inventory space like anything else.
Most items aren't worth re-crafting simply because there are higher-tier upgrades waiting. Re-crafting an iron broadsword if you get awful enchantments isn't a bad idea (if you are going down that road), but once you get beyond iron the expense to craft is typically too high to keep crafting things over and over.
Fortunately, there is an NPC who can help with the latter (the Goblin Tinkerer). For your hard earned gold - "better" items are more expensive - he can re-roll the enchantments on an item. There are some restrictions; there are melee and ranged only enchantments for example. Eventually he can help you perfect your "gear of choice" though. It's generally not worth it to do this except on the very best gear, or the accessories you know you are going to be using a lot. Having something like an Obsidian Shield with +4 defense is gravy.