Though it is not as clear as in previous editions, I believe that you would use the various intelligence skills based on what creature you are trying to learn about.
PHB, page 177 - 178, Intelligence Checks
Your Intelligence (Arcana) check measures your ability to recall lore about ... the inhabitants of planes
Your Intelligence (Nature) check measures your ability to recall lore about .. plants and animals
Your Intelligence (Religion) check measures your ability to recall lore about ... practices of secret cults
I take this as using the specific skills to figure out the in game knowledge. So, for example, I would organize it loosely in the following way:
Arcana - Use this skill to discover more about elemental creatures, creatures of pure magic, arcane creations, and creatures of other planes
History - Use this skill to learn more about creatures that play prominent roles throughout history. For example, goblins, kobolds, and most other humanoid races play significant roles in history.
Nature - Use this skill to learn more about creatures tied directly to nature. Most often, this means animals (wolves, bats, etc.) but it could also be tied to druidic creations, or guardians of nature.
Religion - Use this skill to learn more about creatures of religious creation. Servants of deities, undead, and other holy or unholy beings would be described with this skill.
In terms of setting the DC, use your best judgement based on the typical DC table found on page 174 of the PHB. For example, if you are in a town ravaged by kobolds consistently, it would be fairly easy for someone to know the few different roles they have in battle, so I'd give that a DC of 10.
Whereas if someone wants to know more about unholy creations most people don't see, but are commonly known about, that would be a religion check of 15 for medium difficulty.
The product you are describing is the 4th Edition Dungeon Master's Kit
This was released midway through 4th edition's release and included a book as well as some maps and tokens. Unfortunately it is out of physical print. You can buy the book only as a digital PDF here.
There is no "Dungeon Master's Kit" for 5e that has been released or is scheduled to be released before Christmas 2014.
Keyword is "kit" which generally means a book, some dice, and some tokens/figures to aid in the visual representation of play.
The 5th Edition Dungeon Master's Guide will release shortly
The DMG will be out December 9th. It is simply a book, a big one, covering lots of rules and providing a ton of setting and optional rules for 5th edition.
4th edition and 5th edition are similar but different
Both tell stories about heroic adventurers but the framework they use to tell these stories and engage the players is different. There is a tonal shift in focus for the rules between editions. The 5e DMG is not compatible with the 4e books.
Buying all 3 books will ensure your son and his friends will be able to play the new edition.
If you do pick up the 5e Player's Handbook, Monster Manual, and Dungeon Master's Guide they will have everything they need to play 5th edition.
There are a few official miniatures out for 5th edition currently but they are not available online.
Wizkids partnered with WOTC to make plastic miniatures for 5e, but they seem to be only available in game shops. You may want to check out a local game/hobby shop as they have these or other fantasy miniatures that he could use.
However any miniatures your son already has are perfectly usable with 5th edition.
Best Answer
Yes!MaybeThere was a project in development called Dungeon Scape that would provide character building tools as well as integration with WOTC's published adventure series.
The 4th edition character builder will continue to be a subscription service (independent of Dungeon Scape), and will remain in service as long as it is economically feasible (ie: as long as people keep paying them for it).
DungeonScape was to be a free app (web, android and iOS) with in app purchases to access specific content beyond some basic stuff (likely what comes in BD&D, though that is speculation).
After a beta period where TDT showed off an app with a rather lot of bugs, Dungeonscape was canned.
Wizards has not announced a replacement for Dungeonscape at this time though said in the press release here, that they are committed to providing viable digital tools for 5e.