This is a partial answer, not definitive as I haven't been able to get my hands on a full unabridged translation of the original text of Ferdowsi's Shahnameh.
Firstly, let's note that the era being described in this part of the Shahnameh is an ancient time in Persia before the Islamic conquest, when Zoroastrianism was the dominant and state religion. This is confirmed by noting the many references to Ahriman, Deevs, etc. in the early parts (at least - I haven't read it all yet) of the Shahnameh. So we should bear this in mind when unravelling cultural and especially religious references.
I searched on the internet for seventh sphere zoroastrianism to see if there is any such concept in their religion. One of the search results was a text, Michael Stausberg, "Hell in Zoroastrian History", Numen 56 (2009), pp. 217–253 citing a Zoroastrian priest who said:
Heaven hell depends on your deed. Heaven is your good deeds and hell is your bad deed. There are seven dakhyus [= spheres]. After death, you are assigned a plane and place depending on your actions.
Further searching with the keyword dakhyu led me to more information about Zoroastrian beliefs:
The Dakhyus are wide expanses of inconceivably huge magnitude, and hence referred to in Avesta in language of this material world as "Vouru gaoyaoiti' (Yt. 10.3) literally 'wide pastures'. They are enumerated in Meher Yt; 144 (from upper to lower) as under;
(1) Aiwi; Dakhyu;
(2) Antare; Dakhyu;
(3) Aa; Dakhyu;
(4) Upairi; Dakhyu;
(5) Adairi; Dakhyu;
(6) Pairi; Dakhyu; and
(7) Aipi; Dakhyu;
Our earthy globe which represents the point of the spinning-top mentioned above is mostly located in the Aipi Dakhyu at the bottom, with its (globe's) upper portion in the Zamrir, transitional region between the Aipi Dakhyu below, and Pairi Dakhyu above. The composition of the top three Dakhyus is superethereal, and lower four ethereal, with the earthy globe, of course, material.
All the seven Dakhyus with their respective Zamrirs, transitional regions, make concentric domes, being ranged like the petals of an onion - the smallest (at bottom) enclosed within next larger, that is, the second; that within the third still larger, and so on; but with this distinction that these Dakhyus being ethereal in composition are inter-penetrating at the bottom Zamrir, and mot one above the other as the onion petals are arranged. In other words, these Dakhyus may be described as concentric and geo-centric with dome within dome, the whole Nisti or the Space in its outer aspect resembling a gigantic spinning top, as stated above, the uppermost widest circumference of which being that of the topmost Aiwi Dakhyu spiraling down to the globe of the earth, which represents the lowest point of the spinning top. Thus the dome of the Aipi Dakhyu at the bottom is encaged in that of the Pairi Dakhyu next larger above it (with Zamrir transitional region between) (please see the Chart); then the dome of Pairi Dakhyu contained in that of Adairi Dakhyu still larger; and so on, till the dome of the Antare Dakhyu being contained in that of the top dome of Aiwi Dakhyu.
So one possibility is that "the seventh sphere" could refer to the uppermost of the dakhyus, the voice of the mourning Feridoun rising through all seven of these mystical spheres of existence.
Another option, mentioned already in comments, is the celestial spheres, originating from Ancient Greek models of the universe, but which had made their way to Persian scientists by the time of Ferdowsi's writing. In this case, there are seven "spheres" corresponding to the seven visible "planets" or moving bodies in the sky: the Sun, Moon, Mercury, Venus, Mars, Jupiter, and Saturn. (This set of seven bodies was immensely important for many ancient thinkers, potentially giving rise to the supposed magical significance of the number seven in many cultures and beliefs even to this day.)
