C5 BCE. See generally OCD(4) pp.138-9, under 'Archelaus(1)'; and further below.
The material of the first and principal part of the present entry comes largely from Diogenes Laertius 2.16.
[1] Or 'called a natural philosopher by choice.' At any rate, see again under delta 1080.
[2] Unlikely. DL loc.cit. does say this (adding that physical philosophy came to an end with Archelaos upon the shift of emphasis in philosophy toward ethics imposed by Socrates); but Clement of Alexandria construed the statement as applying not to Archelaos but to Anaxagoras, and the Loeb editor of DL, R.D. Hicks, translates accordingly.
[3] That Archelaos was a mentor of Socrates is asserted by Aristoxenus and Theophrastus as well as by DL. The association of Archelaos, Anaxagoras, and/or Socrates with the education of Euripides, however, strains chronology perhaps too much.
[4] Nothing of Archelaos' writings survive except one fragment, "coldness is the connection" (presumably which holds the Earth in a solid state); see Diels and Kranz, Die Fragmente der Vorsokratiker, 60B1a. From surviving testimonia (principally Hippolytus), he would seem to have held generally to the principles of Anaxagoras (see alpha 1981). He posited a geocentric universe, governed by the interplay of the hot and the cold, and the evolution of life from a primordial sludge through civilized society with "mind" as a guide. Compare the anthropology of Plato in the Protagoras.
[5] This addendum (absent from most of the manuscripts) comes from beta 152, and has been added simply because of the coincidence of the name Archelaos. (Note that presbu/tou in beta 152 has become presbute/rou here.)