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).

Yes, it does: a Mirror of Suggestion:

This small hand mirror manifests its power up to twice per day against creatures that see their reflections within it, as a gaze attack with a range of 30 feet. Potential victims can attempt to avert their eyes or look away as they would from a normal gaze attack. A creature meeting its own gaze in the mirror must succeed on a DC 13 Will save or be affected by a suggestion chosen by the mirror’s wielder.

The suggestion encoded is "I have a right to be here as a duly appointed representative of your superiors. Tell me who I am."

Or, much the same can be done with a potion of glibness, as in fantasy settings, authority isn't vested in paper, but in recognized people.

Or, barring that, you could be a level 1 binder (on top of whatever else you were, probably factotum/bard) having bound naberius, and just use disguise self to disguise yourself as someone who is appropriate to be there. (I note naberius qua binder rather than a normal spellcaster because a binder can keep this charade up all day at very low levels without excessive resource use.)

Or, barring that, you could be a diplomancer and just cause whomever you speak with to become fanatically aligned with your interests.

Or, barring that, have a piece of paper with "I prepared explosive runes today" written on it. It's the ultimate password.