Does Allah give the human population exclusive right, to kill and eat any other baser animal life forms?
Also, a corollary, is it immoral to be a vegetarian and not eat killed animals in Islam?
I don't know what "exclusive right" in the question precisely means, but Allah created other hunting, meat-eating animals besides humans.
So I think it's safe to say humans don't have exclusive rights to killing and eating animals.
It's well-known that Muslims are forbidden from eating meat from pigs, so clearly we can't eat all meats. There's other instances of meat being forbidden in the Qur'an:
Qur'an 2:173 states:
He has only forbidden to you dead animals, blood, the flesh of swine, and that which has been dedicated to other than Allah. But whoever is forced [by necessity], neither desiring [it] nor transgressing [its limit], there is no sin upon him. Indeed, Allah is Forgiving and Merciful.
While eating meat is explicitly made permissible in the Qur'an, being a vegetarian is also permissible; it's neither encouraged nor discouraged.
Mufti Ikram ul Haq writes:
Therefore, if someone just doesn't want to eat meat and instead chooses to eat only vegetarian or vegan food then there is nothing wrong but if someone starts to believe that the meat of those animals which are Halal is not Halal or Haram then this will be a grave mistake and error of part of faith.
To kill for recreation is either forbidden or discouraged. Islam Q&A talk about hunting and fishing for sport:
If the hunter or fisherman ... is only doing it for sport ... then the ruling on hunting or fishing in this case changes from permissible to disliked (makrooh).
Basically, it is Haram to make an animal or bird a shooting target.
An MuftiOnline fatwa agrees with this.
The answer to Recreational killing of animal gives several ahadith about kindness to animals. E.g.
Whoever kills a small bird for no reason, it will beseech Allah on the Day of Resurrection saying: O Lord, so and so killed me for no reason. And he did not kill me for any beneficial purpose. -- Sunan an-Nasa'i 4446 [grade: hasan] (sunnah.com)
Islam says eat the herbivorous animals and not the carnivorous. Also, sea foods are made Halal(allowable). And also,
Prohibited to you are dead animals, blood, the flesh of swine, and that which has been dedicated to other than Allah , and [those animals] killed by strangling or by a violent blow or by a head-long fall or by the goring of horns, and those from which a wild animal has eaten, except what you [are able to] slaughter [before its death], and those which are sacrificed on stone altars, and [prohibited is] that you seek decision through divining arrows. That is grave disobedience. This day those who disbelieve have despaired of [defeating] your religion; so fear them not, but fear Me. This day I have perfected for you your religion and completed My favor upon you and have approved for you Islam as religion. But whoever is forced by severe hunger with no inclination to sin - then indeed, Allah is Forgiving and Merciful. (Qur'an 5:3)
is it immoral to be a vegetarian and not eat killed animals in Islam?
Doctors say that it is highly advisable to have green leafy vegetables. But if a person doesn't like to eat them, is he being immoral?
Similarly, Allah says, that meat has a nutritious value for humans, then if someone Muslim prefers to be a vegetarian for any reason is not being immoral.
Meat is rich in protein, iron, vitamin B1 and niacin. Non-vegetarian food is a good source of excellent protein.
You're curious by thinking how Islam justifies eating animals and if you wish to get you're questions answered, like...
Hindu scriptures give permission to have non-veg
It is not compulsory for a Muslim to have non-vegetarian food
Then this article is especially for you...
May the creator guide us all.