This shloka is also called as the Pratignā shloka as the complete Gita teaching is presented in this shloka.

In the whole world the one consciousness principle is the only thing that is indestructible. All other things changes or gets destructed.

Bhagavān Krishna explained that the Ātma, one consciousness principle is not destructible thus it existed from the beginning. So it was not born, thus which was not born can not be killed. Therefore he was not killing anyone and so he can be free from that sorrow.

Being a Kshatriya, Arjuna's main duty was to stop Adharmis and uphold Dharma and Justice. So this war was necessary for the upholding of Dharma.

Ātma is
Nityaḥ - Always present.
Nirvikaraḥ - Which cannot be modified.
Sarvagathaḥ - Present everywhere.
Akartā - Not the doer.
Abhokhthā - Not the enjoyer.

Ātma is also called Aprameyaḥ as it is not the object of knowledge, it is self evident. It existed in the past, present and will be present in the future.

The body undergoes different stages of modification thus forming the cycle of life. These stages in the proper order are:

Asti - Existence
Jāyate - Birth
Vardhate - Growth
Vipariṇamate - Transformation
Apakṣīyate - Decline
Vinaśyati - Death

No, Ātma is Nirvikaraḥ which means it does not undergo any changes and is present always (Nityaḥ)

The body undergoes different stages. The proper order is:

Kaumāraṁ - Childhood.
Yauvanaṁ - Youth.
Jarā - Old Age.

This implies that the soul:

Cannot be seen, touched, or measured like physical objects
Cannot be analyzed by scientific instruments or experiments
Exists beyond the limitations of human senses and mind
Is eternal and unchanging, unlike the body