Do artifacts have summoning sickness?
Like to use its ability, such as elixer of immortality.
Artifact creatures do, but regular artifacts do not, unless stated otherwise.
technically everything has summoning sickness but it only affects creatures.
Actually not, only creatures have summoning sickness. Summoning sickness means you can't attack or tap creatures without haste for triggered abilities with the {T} symbol or the "tap this creature: do something" effect if you haven't controlled them since the beggining of your upkeep. (if a card says, say, "tap five untapped Myrs you control. {Myr battlesphere} gets +x/0 and deals x damage to defending player", it doesnt count to sickness)
yes but a {glint hawk idol} which is animated can't attack the first turn it is in play
For each creature, you can't attack with it or use its {T} abilities unless you controlled it since the beginning of your turn.
Say you cast a {Chalice of Life}. You could tap it right away if you want, but if you made it a creature with {Tezzeret, Agent of Bolas}' -1 ability, then you couldn't tap it for its ability.