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.
No, this does not exist in 5e.
Implementing it in 5e may be less satisfying, since the increase in proficiency bonus covers many of the incremental advancement options from previous rulesets in one swoop.
So for the fighter, for example, you'd have two options - the one feature they get every level or the proficiency advance they get every 5th level. It's pretty difficult to make that incremental. Even with something like a druid, they get a spell advance and then 0-2 other improvements. Within a single level there's not enough advance, on average, to break it up much.
Another way to implement incremental advancement is to let the character apply the next larger proficiency bonus to one of (attacks, saves, skill checks) per level they go up (not between levels). That seems pretty fiddly, though, and lets someone min-max what they're already good at levels ahead of time.
Best Answer
10 yes (sort of), 20 no.
Taking 20 was not a thing in 4th edition, and is similarly absent from 5th. Taking 10 however is sort of a thing in 5th, but it's not implemented the way you might think it should be.
Basically, the way "taking 10" works in 5e is that every ability (And by extension, skill), has a "passive" score (Basic Rules v2 page 59). This is 10 + modifier, and it sort of represents your natural, not at all under pressure ability in a specific discipline.
The best example of this is Passive Perception. Basically, if you walk into a room, your passive perception is what you instantly notice. Many items in 5e have a higher DC if someone is passively looking for it than when they are actively doing so. But Passive Perception doesn't draw attention that actively looking might.
Ultimately, when it comes to taking 20, this gets back to a fundamental D&D principle. If failure isn't interesting on a specific roll, there is no sense in rolling the check at all. This is the problem that take 20 solved, and while 5e could fall victim to it, I've found in practice that it really doesn't. I've run several sessions where there was no need to roll dice at all, as situations pretty much just were RP and the characters often found automatic success on things like Diplomacy or bluff etc because their passive scores are high, but also because the DCs were really low for things like that (what can I say, folks in Phandalin aren't shy when it comes to telling PCs stuff).