Making a distinction between a mortal thing and to annihilate something.
With the former, a mortal thing can be killed, made to die, die, with the latter, an annihilated thing is reduced to nothing.
From a Catholic perspective, to create means to call [a being] into existence out of nothing.
CCC IN BRIEF
No creature has the infinite power necessary to "create" in the proper
sense of the word, that is, to produce and give being to that which
had in no way possessed it (to call into existence "out of
nothing") (cf DS 3624).
Thus the reverse action, the opposite of create is annihilate and not kill/make to die.
ALL creation, the human soul included, depends on God for its conservation or continuance in being.1
CCC IN BRIEF
God created the universe and keeps it in existence by his Word, the
Son "upholding the universe by his word of power" (Heb 1:3), and by
his Creator Spirit, the giver of life.
Thus from Catholic teaching and Sacred Scripture, ALL creation, the human soul included, would go back to the nothingness from which it was called if God so wished and withdrew his upholding power.
1. cf. God upholds and sustains creation, CCC 301 and Relation of God to the Universe | New Advent.
The soul, as the OP pointed out, is immortal, and made so from the very first moment of its existence when created by God. Thus the immortality of the soul is a natural property of the soul in the image of its Creator, again as correctly stated by the OP.
What is the answer if the OP's question is understood as, [I]f God created the soul immortal, can God kill it?
This is would be a classic contradiction and better dealt with by philosophers. The argument against this scripturally would be:
God is not man, that he should lie,
or a son of man, that he should repent. Has he said, and will he not do it?
Or has he spoken, and will he not fulfil it?
Addressing the OP's difficulty that led to the question
Will souls - accurately the whole person, body and soul - be tormented forever or be annihilated?
I am not sure I can present another answer if this one does not suffice.
If God so loved the world that he gave his only Son, that whoever believes in him should not perish but have eternal life, then the reasoning is faulty that projects God 'as unfair, mean, sadistic, to torture' souls forever in hell.
Quite frankly, the inspiration behind those wanting personal annihilation should they of their own volition fail to attain eternal life, is the same inspiration, barring illness, that's behind suicide. Akin to 'taking the ball and going home'.
What about the second death that scripture speaks about?
The linked passage above equates that to [being thrown into] 'the lake of fire' and 'one's name not being written in the book of life' i.e. not having 'eternal life'.
While angels and human souls - rational spiritual beings - have a natural immortality, God called them - angels and men - and willed that they, cooperating with him, would attain to supernatural eternal life, i.e. share in his own life, drawing from it. The absence of this life is the second and eternal death.