There is a real United Nations Office for Outer Space Affairs (UNOOSA), but it is the United Nations office responsible for promoting international cooperation in the peaceful uses of outer space, not one dedicated to contact with extraterrestrial life.
Avogadro's constant, displayed on Dave's tie clip, is a number: 602,214,076,000,000,000,000,000 to be exact. Dave states that it helps calculate the number of atoms in a substance, but this isn't entirely correct: it calculates the number of particles in a given substance, which, depending on the substance, can be atoms but also entire molecules.
Amy's apartment has the number 314, the first three digits of Pi, which equals 3.14159265359... which Sheldon knows to its 1000th digit (and can belch to 21 digits without vomiting).
This episode was taped on November 10, 2015.