Can the Soul take birth in higher planes or lower planes?
The soul is never born and it never dies. The soul is indivisible, eternal and invincible.
Sri Krishna says in Gita 2:20
The soul never takes birth and never dies at any time nor does it come into being again when the body is created. The soul is birthless, eternal, imperishable and timeless and is never destroyed when the body is destroyed.
Then who is it that takes birth?
It is the body takes birth and not the soul. As you have quoted already that
Sri Krishna says in Gita 2:22
Just as a man giving up old worn out garments accepts other new apparel, in the same way the embodied soul giving up old and worn out bodies verily accepts new bodies.
Then who is enjoying and suffering in the various planes of existence?
The one who is in ignorance is the one who suffers. The one who thinks He is the body, that person is the one who suffers in all the worlds.
Sri Krishna says in Gita 2:71
That person attains peace who giving up all material desires for sense gratification lives free from attachment, free from false ego and sense of proprietorship.
So basically in other words, the one who has realized that He is the Soul, knows that all suffering DOES NOT HARM him in any way. All the sufferings are only at body-level and the soul is unaffected.
But the one in ignorance, who thinks that he is the body, undergoes various suffering, because he is mistaken his true identity. He seeks sense gratification, because sense gratification pleases the body and He thinks He is the body. Thus in His endeavour for sense gratification, He is put through several turmoils due to the working of the various gunas.
Sri Krishna says in Gita 2:42,43
O Arjuna, men of limited understanding presume speculative interpretations of the Vedic scriptures, advocating that there is no divine principle in creation; full of lascivious desires, aiming to attain the lush heavnly worlds; they glorify only the statements in the Vedas which are pleasing to their senses; performing numerous ostentatious rituals productive of good birth, wealth and power insuring sense enjoyment and worldly pleasures
Thus they suffer because they think they are the body, and not the soul.
All the best!