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Are there any class, proficiency, or skill requirements 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.

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There are two official optional rulesets for crafting

Both are optional and require that you work with your DM as to if they're an option and how they'd work. In either place that these rules appear, they are optional variants to the base rules.

Dungeon Master's Guide

The first official optional rules released appear in the Dungeon Master's Guide in chapter 6 as part of the "Downtime Activities" section:

Crafting a Magic Item

Magic items are the DM’s purview, so you decide how they fall into the party’s possession. As an option, you can allow player characters to craft magic items.

Within these rules, there are the following requirements:

  • a formula that describes the construction of the item
  • the character must also be a spellcaster with spell slots and must be able to cast any spells that the item can produce (and expends the relevant spell cost each day during crafting)
  • the character must meet a level minimum determined by the item’s rarity
  • a required gold and time cost to craft the item during downtime (consumable cost half as much as other magic items)

\begin{array}{c|c|c} \text{Rarity}&\text{Cost}&\text{Minimum Level}\\ \hline \text{Common}&\text{100 gp}&\text{3}\\ \text{Uncommon}&\text{500 gp}&\text{3}\\ \text{Rare}&\text{5,000 gp}&\text{6}\\ \text{Very Rare}&\text{50,000 gp}&\text{11}\\ \text{Legendary}&\text{500,000 gp}&\text{17}\\ \end{array}

Xanathar's Guide to Everything

An alternative ruleset is part of the optional variant to downtime as a whole (and one I personally think is more fleshed out) that appears in the supplement Xanathar's Guide to Everything in the "Downtime Revisited" section of chapter 2:

Crafting Magic Items. Creating a magic item requires more than just time, effort, and materials. It is a long-term process that involves one or more adventures to track down rare materials and the lore needed to create the item.

Within these rules, there are the following requirements:

  • a formula for a magic item in order to create it
  • an exotic material to complete it. This material can range from the skin of a yeti to a vial of water taken from a whirlpool on the Elemental Plane of Water. Finding that material should take place as part of an adventure. (the suggested challenge rating of the creature to face, though not necessarily harvest the material from, appears in the table below)
  • whatever tool proficiency is appropriate, as for crafting a nonmagical object, or proficiency in the Arcana skill.
  • a required gold and time cost to craft the item during downtime (consumable cost half as much as other magic items). Do note that downtime rules as a whole are revised when using the Xanathar's Guide to Everything rules

\begin{array}{c|c|c|c} \text{Rarity}&\text{Challenge Rating}&\text{Time}&\text{Cost}\\ \hline \text{Common}&\text{1-3}&\text{1 workweek}&\text{50 gp}\\ \text{Uncommon}&\text{4-8}&\text{2 workweeks}&\text{200 gp}\\ \text{Rare}&\text{9-12}&\text{10 workweeks}&\text{2,000 gp}\\ \text{Very Rare}&\text{13-18}&\text{25 workweeks}&\text{20,000 gp}\\ \text{Legendary}&\text{19+}&\text{50 workweeks}&\text{100,000 gp}\\ \end{array}

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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 ...

emphasis mine

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.

Adventurer's League

In the AL, you will not be able to craft anything other than healing potions. All other forms of magic item creation are prohibited.

Xanathar's Guide to Everything

Since much has changed since I answered this question, Xanathar's Guide to Everything has an optional replacement for the Item Creation system in the Dungeon Master's Guide. This system has rules that have vastly expanded upon and, if used, may change the specific requirements for a given spell.

Artificers in Eberron: Rising From the Last War###

While this doesn't directly answer the question itself, it is noteworthy that Eberron: Rising From the Last War includes a class that is centered around item creation, the Artificer. The rules for this class are very different from other examples of item creation and are precise on their requirement and discusses using narrative flavor and working with the DM in order to add a techno-magical, alchemical, or otherwise mystical creator based feel for your spells.

The class does not allow free creation of unlimited magical items, but creating items and bonding with them are related to their class features.

Consumables

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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  • \$\begingroup\$ does the spellcaster restriction apply to magic items that do not require a specific spell to be cast though? I think a sword +1 can be just crafted by a dwarven smith without having him to become an eldritch knight :) \$\endgroup\$ – Ghilteras Feb 23 '17 at 19:45
  • \$\begingroup\$ By default, yes. The basic requirements are formula and spellcaster with spell slots (i.e. not a warlock). If the item can produce any spells, you then need to be able to cast those spells as well. \$\endgroup\$ – Aviose Feb 24 '17 at 14:51
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    \$\begingroup\$ I would add that in the answer (in AL only healing potions are craftable). Also it would be worth pointing out that consumables (like spell scrolls and potions) require half cost and downtime. \$\endgroup\$ – Ghilteras Jun 22 '17 at 16:59
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    \$\begingroup\$ @GhilterasAdded my last comment to you, a section on UA, AL, and consumables. \$\endgroup\$ – Aviose Jun 22 '17 at 17:50
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    \$\begingroup\$ @Aviose Warlocks have spell slots and can create magic items. A fighter with the Magic Initiate feat is an example of a spellcaster that doesn't have spell slots. \$\endgroup\$ – Derek Stucki Jun 22 '17 at 19:31

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