According to Bhagavad Gita, Moksha (liberation) and Amaratva (immortality) are same.
From this answer, for liberation & immortality -- happiness-Sorrow or any duality is same.
BG 2.15 - O foremost of men, surely all these don't distress to that person(Purusha); that wise person to whom sorrow and happhiness are the same -- is fit for immortality.
BG 6.7 - The conqueror of the own self, quite pacified [person] establishes with supreme Self (viz. liberates) in heat-cold, sorrow-happiness, and honour-dishonour
Besides, you may also combine below verses to confirm that both are same.
BG 8.16 - All planes along with BrahM-plane are repetitive, O Arjuna. But O son of Kunti, reaching "Me", the rebirth doesn't happen.
BG 9.19 - O Arjuna, I give heat, I withhold and pour the rains. the immortality is "Me", and also death, existent and nonexistent.
The meanings of liberation and immortality are also combined in a verse:
BG 2.20 - "That" is neither born nor die; Nor having come to exist, "That" will again cease to be. "That" is birthless, eternal, undecaying, ancient; It is not killed when the body is killed.
"Moksha - Once you are dead you'll get rid of Life Cycle (Jeevan Chakra), you become an absolute and pure energy traveling across the Universe"
No, not quite true. If after liberation one "becomes" an energy of whatever form, then that means it's still existing. In other words, that entity has taken a rebirth in form of an energy.
Any existence is directly orthogonal to liberation. :-)
"Amarta (Immortality): You remain young forever. You will never die."
No. As already stated above, those who are born are meant to decay followed by death and those who die are meant to take rebirth.
BG 2.27 - Those who have a birth, it's certain [for them] to die, For dead, certainly there will be a [re]birth; Hence you don't have to grieve about inevitable
Until a being relates themselves with bodily aspects, they keep taking births.
"Lord Hanuman, who is Lord Shiva's avatar is immortal, why he didn't try for Moksha?"
No. Lord Hanuman is not immortal. He is a Cheeranjivi (long lived). There are various posts on that topic in our site.
Moksha is not something to be tried or achieved, it's received or attained, once a person becomes totally indifferent to everything in the worldly aspects. Until the gods like Shri Hanuman have certain role to play or duty to perform, they may not give up their physical self.
"Should we pray for Moksha or Immortality?"
None. Anyways both are same.
If we pray to get "something", then we have to stay till that "something" is attained. Hence, until we have some purpose to fulfil, Moksha is not attainable.
Desires (viz. to achieve something) & Anti-desires (viz. to give up something), both keep us plugged with the material world in one or the other form.
Disinterest or indifference, eventually leads to liberation.
Refer this answer.