The short version is that we don't know; both are possible, but there's no evidence one way or the other.
Did Círdan look into the Stone of Elostirion?
Possibly. We know that other Elves did so on occasion, as Tolkien discusses in The Road Goes Ever On, as noted by Hammond and Scull in A Reader's Companion, regarding the company of Gildor Inglorion (met by Frodo et al on the way to Bree in Fellowship):
[S]ince they appear to have been going eastward, [they] were Elves living in or near Rivendell returning from the palantír of the Tower Hills. On such visits they were sometimes rewarded by a vision, clear but remote, of Elbereth, as a majestic figure, shining white, standing upon the mountain Oiolosse
The Lord of the Rings: A Reader's Companion Book I Chapter 3: "Three is Company"
Although we don't have direct confirmation that he ever did use the Stone, likewise we have no reason to believe he didn't. It could go either way.
Did Círdan use the Stone to communicate with the West?
Possibly, but we don't know. We do know that the palantíri weren't capable of communicating except with other palantíri; you couldn't use one to beam your thoughts into the head of a random schmuck:
The palantíri could not themselves survey men's minds, at unawares or unwilling; for the transference of thought depended on the wills of the user on either side, and thought (received as speech) was only transmittable by one Stone to another in accord.
Unfinished Tales Part 4 Chapter 3: "The Palantíri"
So it could only communicate with the Undying Lands if there were another Stone there; there is evidence for such a thing:
[I]t is believed that thus [Elendil] would at whiles see far away even the Tower of Avallónë upon Eressëa, where the Masterstone abode, and yet abides.
The Silmarillion V Of the Rings of Power and the Third Age
But we don't know if the Master stone of Avallónë was capable of receiving messages or not; we certainly never hear of any being passed.