Are there any class, proficiency, or skill requirement to craft a magic item in D&D 5th edition other than to possess the minimum character levels?
I.e. 3rd level for Common and Uncommon Magic Items, 17th level for Legendary Items.
Yes, there are rules listed for crafting magic items in the DMG on pages 128-129. A list of capabilities, minimum levels, creation costs in gold pieces, and other requirements is listed there. The time-frame to create is also listed with these details.
I will give one example: A common item would require a formula, 100 gp in components, and a minimum of a 3rd level spell caster with a similar spell.
Due to requests for a specific part of this information, I'll quote the second paragraph of the DMG on Crafting a Magic Item:
... To start, a character must have a formula that describes the construction of the item. Their character must also be a spellcaster with spell slots and must be able to cast any spells that the item can produce. Moreover, the character must meet a level minimum determined by the item's rarity, as shown in ...
That said, the DMG states that these are, as with all things in the DMG, options. They are the specific guidelines, so they function as RAW, however, house rules are emphasized as possible, as is not allowing players to craft them at all.
Outside of environments like the AL, ensure you discuss this with your DM. Most would reasonably pursue pathways to make at least basic magic items (like a +1 weapon) for a non-caster if the items they are made of are rare/magical enough.
The WOTC team released a playtest version of new/expanded Downtime rules that included item creation. It includes a section on magic items that is worth perusing.
As with all UA material, it is playtest, may not be completely balanced, and is not Adventurer's League legal, but may be officially released in a later D&D5e book.
In the AL, you will not be able to craft anything other than healing potions. All other forms of magic item creation are prohibitted.
Consumable magic items cost half the amount of time and resources of non-consumable items of the same rarity to create.
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