"What is the root meaning of "soul" in this verse?"
The Hebrew written here [נפשׁי - nun, pey, shin, yud] translated as "soul" is nepheshi, indicating "my soul". The stem word being nephesh [as a substitute for a pronoun, nephesh נפשׁ is often is written with the pronominal suffix yud - as נפשׁי ].
Nephesh is the parent noun from which naphash is derived - that verb form meaning very simply "to take breath" or "to refresh oneself". The original, Hebrew concrete meaning of the word, is simply to breath. Not the breath itself, but rather the act of breathing - specifically, a creature who breathes.
Nephesh is a word which occurs quite early in the biblical texts. "God said, "Let the waters [hayamin] teem with teeming [sherets] living [chayah] creatures [nephesh] ..." - Genesis 1:20, and just following, "And God thus created the great sea-giants and every living [hachayah] being [nephesh] that creeps, with which the waters teemed after their kinds ..."
Creatures who breathe. It was later used in Genesis to describe also what man came to be - a creature who breathes.
In its paleo concrete sense it pictures the function of "reproduction or seed bearing [from the pictograph of the nun] creature or being [from the shin, the two front teeth] who breathes [from the pey, to blow]". Used first of animals and living things not "human" brought forth from the water, which then reproduced, and filled the entire earth, and the air above. Brought forth in the fifth creative-period [or yom], long before "man" enters the scene in the sixth creative-period [or yom]. Man followed animal and other living beings.
It's verb form means to refresh - just as a breathing a few deep breaths by a creature or being who breaths can refresh the life which exists in the blood. Its verb form occurs in Exodus 23:12, "You shall do your work six days, and on the seventh day you shall rest, so that your ox and your ass may rest, and the son of your slave-girl and your alien may be refreshed [naphash]. [LITV]. In its later biblical uses it indicates hunger, desire, emotions - making nephesh a being's [animal or human or creature] very life existence - the whole sum "a life".
"Is it the same as "spirit"?
No, the Hebrew word most often translated by the English word "spirit", is ruach, which is a word concretely meaning an unseen, impersonal force which enacts upon other things. Like the wind causes the leaves of trees to flutter and move about, or like how electricity makes a blender swirl and function.
The KJV version renders Amos 4:13 as "For, lo, he that formeth the mountains, and createth the wind [ruach], and declareth unto man [...]" The first use of the Hebrew word ruach is in Genesis 1:2, "the earth being unformed and void, with darkness over the surface of the deep and a wind [ruach] from God sweeping over the water" [Oxford JPS TNK, Jewish Study Bible].
There is no place in any of the Hebrew or Christian Greek biblical texts wherein nephesh and ruach are used in synonymous parallel.
"Is it the same as "life-force"?"
No. Nephesh is manifested, that is, made to be living, as a result of the application of "nishmat-ruach" to flesh and blood according to the bible. All nephesh [animal, man, all cratures who breath] are said to share this same life force, and this same living spirit, which is from God and not a possession of the living being.
"And all flesh died that moved upon the earth, both of fowl, and of cattle, and of beast, and of every creeping thing that creepeth upon the earth, and every man. ALL in whose nostrils was the "breath of life" ..." - Genesis 7:22, KJV.
The "breath of life" - "nishmat-ruach chayim", is asserted to be in all of these things. It is therefore not unique to man, and man alone. Elihu says of his God Yhwh, "If that one's spirit [ruach] and breath [neshamah] he would gather to himself, ALL flesh would expire [breath-out] together, and earthling man himself would return to the dust [Job 34:14-15].
Some claim that nephesh is the soul of man which transcends or survives death - yet the bible shows that nephesh dies, and that nephesh is all "creatures who breath"; animal and beast first, humans following. So some claim that it is ruach, specifically nishmat-ruach "the breath of life" which the bible says is unique to man, and survives death as being immortal. Yet the bible itself denies this thought in its very texts - the "spirit' belongs to Yhwh, it is his, it exists in all living beings who breath, and if he should take it back from them, they perish, they die, they no longer live.
"Only, God doth ransom my soul from the hand of Sheol, For He doth receive me" - he provides a ransom, to buy back your life existence for you from the common grave of all mankind - i.e., death. A resurrection - from nonexistence to existence; from nothing to something.
"perhaps nishmat is some strange declension of nephesh. Is this so? If not, then what is the origin of the third term?"
Nishmat is actually a form [declined if you will] of neshamah. Neshamah is a noun derivative [form of] nasham [the "third term"]. Nasham means literally "to pant" [Brown Driver Briggs]. Nasham in its stem form occurs only in Isaiah 42:14, "I have long time holden my peace; I have been still, and refrained myself: now will I cry out like a travailing woman; I will gasp and pant [nesham -נשׁם] together." [ASB, but reads nearly identical in the WEB, ERV, etc].
This noun derivative [neshamah] when used in reference to man or animal generally signifies the breath of the force of life, as in Genesis 7:22 as cited. The Hebrew which transliterates as "kol asher nishmat [breath] ruach [unseen impersonal force] chayim [life] be'apav mikol asher be'charavah metu" can be read as "everything on dry land whose life was sustained by breathing died" [JPS TNK]. Everett Fox's translation "The Five Books of Moses" equally reads "all that had the breath of the rush of life in their nostrils, all that were on firm ground, died".
The only other derivative of nasham is tinshemet, meaning "an animal". Nephesh in its basic Hebrew concrete meaning indicates a creature [of procreation] who breathes [neshamah]. Neshamah, a form of nasham, is used synonomously [in virtual] with, nephesh [you might wish to compare such use in De 20:16; Jos 10:39-40; Jos 11:11; 1 Kings 15:29, and so on] as well as the parallel in Genesis 2:7 "And the LORD [Yhwh] God formed man of the dust of the ground and breathed into his nostrils the breath [nishmat] of life [chayim]; and man [ha'adam] became a living [chayah] soul [nephesh]". [KJV].
The record at Genesis 2:7 uses nishmat [declined neshamah] in describing God's causing of the man's body to have animated life so that he became "a living being/creature who breathes".
The TDNT offers the following "Breath may be discerned only in movement [as in the movement of the chest or expanding of the nostrils], and it is also a sign, condition and agent of life, which seems to be especially tied up with breathing". Neshamah/nishmat or "breath" is both the product of the ruach or life, or force of life, and also a principle means [as Thomas Paine in his post above so also notes] of sustaining that force [ruach] of life [chayah] in living creatures who breath [nephesh].
Science might state it similarly as a force of life which is present in every living cell of the body's trillions of cells, and that while millions of said cells may die each minute, constant reproduction [hence the Hebrew's use of the nun in nephesh] of new living cells goes on simultaneously. The continual perpetuation of the force [ruach] of life [chayah] in a living body, be it animal or man or fish or any other living creature who breathes, is dependent upon the air/oxygen which the act of breathing brings into the body. Stop the breathing [neshamah or nasham] and the blood runs out of oxygen, and the creature who breathes [nephesh] will eventually die and cease to exist. Likewise, should God remove his ruach [as mentioned in the former cited biblical verse in Job] and his breath [neshamah or nasham] all living creatures who breath [nephesh] would die. You might find Genesis 9:4 interesting in this regard.
Even in our English vernacular, breathing is nearly inseparably connected with life, so is nephesh impossible absent of neshamah/nasham, and so is neshamah/nasham impossible without God's giving of his ruach.