[Edit]: Parts of this answer respond to removed content in older versions of the OP, and to comments. The current version of the OP deserves some elaboration of this answer. (And, IMHO, other answers too!)
There are other spiritual "worlds" than those that are dualistic. By common psychological definitions of spirituality, the existence of an "ethereal soul" is not a definitional belief. That is not to say the "spiritual world[s]" to which you're referring are invalid—only that spirituality does not necessarily require belief in anything supernatural or unobservable. Psychological theory on spirituality is quite diffuse itself; in my own study of it (it's among my top five research interests!), I've found it hard to really limit what kinds of belief systems qualify as spiritual, except in as much as they entail some sense of self-transcendence, interconnectedness among identities of discrete beings, or at least an opposition to reductionistic or nihilistic systems that ascribe mundanity or emptiness to phenomena in general.
This is not to say that spirituality itself (i.e., spiritual beliefs) can't be studied scientifically either; in fact, some admirable progress has been made in defining a measurable set of dimensions by which spiritual beliefs can be assessed and compared at the individual level. Ralph Piedmont, with whom one of my former labmates has worked directly, has made a lot of this progress himself, so I'd recommend checking out some of his work to get a sense of what research on spirituality is like in the professional psychological community. He's even proposed spirituality as a "sixth major dimension of personality" that belongs in the same general domain of individual differences in personality traits as the Big Five Factor Model, and he's reported success in generalizing his operationalization of spirituality cross-culturally (Piedmont, 1999, 2007; Piedmont & Leach, 2002).
As I responded to your comment, "The oneness that we all share on accord of having the same Godly element within us" is also not a definitional belief in all forms of spirituality. You'll see this if you consider Piedmont's definition of spirituality (and don't find it too vague to be valid). Here's a brief quote from Piedmont (2007) that I think sums it up about as succinctly as any other quote from him that I've seen:
To capture a broad-based aspect of spirituality that would be nondenominational in nature, Piedmont (1999)...Working from a trait perspective for construing spirituality...operationalized spirituality as a motivational drive directed towards creating personal meaning for one’s life. Three correlated subscales were identified: Universality, a belief in the unitive nature of life; Prayer Fulfillment, a feeling of joy and contentment that results from personal encounters with a transcendent reality; and Connectedness, a belief that one is part of a larger human reality that cuts across generations and groups.
If you look further into his work, you may note that the "unitive nature of life" is not required of all forms of spirituality. Universality is measured as one of three dimensional aspects, any one of which may be sufficient alone to qualify a belief system as spiritual (or at least, not completely non-spiritual) wherever present. Nor does Universality necessarily involve the existence of any "Godly element" anywhere. Even to whatever extent people believe we share oneness through having any of the same elements we have within us, Connectedness does not necessarily involve anything supernatural; definitionally (as in the above quote), it does not involve much more than "a larger human reality."
For reasons such as these, it is necessary that you narrow down the concept of a soul before it can find any scientific support. Consider your statement, "Everything that we feel or do or think is attributed to our soul." This conflicts with most folk and religious concepts of a soul with which I am familiar, at least when taken literally (your intended meaning may be necessary to specify here as well, if you didn't mean this literally). Most concepts of a soul do entail some amount of dualistic belief, whereby some feelings/thoughts are attributed not to the soul, but to the brain, mind, or body. (Even these three entities are not necessarily the same thing either, depending on whom you're talking to!) This lack of consensus on basic, crucial definitions is partly what makes the issue so difficult to separate the scientific issue from philosophical or religious issues.
If you can narrow it down, we might get somewhere, if only (I'd bet) to conclude that your definition of an ethereal soul is fundamentally inaccessible to empirical science for lack of measurability or falsifiability. Without narrowing down the concept of what would qualify as a soul more specifically than you have in your question, I suspect someone could come up with an equivalently strong counterargument to any scientific support one could give for an ethereal soul, so long as ethereal means unobservable and unknowable in some sense.
One complication that remains in the updated OP so far is that biological, cognitive, or social explanations of sentiments don't necessarily preclude a loosely defined "soul" from adding explanatory power. These three kinds of scientific explanations often work together (and may not even be distinct from one another, ultimately) in our models of behavior, emotion, and cognition; I'd say there's still plenty of room for more explanatory factors. The hard part is finding others that actually provide falsifiable explanatory power in ways that existing scientific theories haven't covered already. I'm unaware of any way of measuring unique, useful information about the soul, so I'd say the prospect of doing so is pretty dismal, even though I can't rule it out entirely (and I don't see how others could either).
In your examples of soul-relevant beliefs, I think this becomes even clearer. Each may deserve its own separate question and answers, but I'll respond very briefly to each here to help you decide whether you want to pursue them.
First, the soul mate concept is very loosely defined itself, and most plausible folk concepts don't seem universally applicable. Lots of research has examined romantic attraction and long-term commitment, and some very mundane factors (e.g., proximity, children) have a lot more influence than one might expect if people were routinely finding their soul mates in an idealized sense. Somewhat mundane factors like communication skills, shared values, and physical attraction seem to play predominant roles even in a more limited, rare sense of "true love". There's surely more to it, but good luck improving any statistical models with a soul concept.
Second and third, the soul may ultimately be little more than a popular metaphor and folk concept of a person's ethical systems, impulse regulation, and psychological well-being. It certainly may not too, because it refers to something supernatural and apparently unfalsifiable; i.e., there may be much more to the soul than we'll ever know in this life (regardless of whether there are other lives). In either case, the challenge remains to distinguish the soul's role in moral behavior from the roles of factors that fall within the scope of scientific research, such as attitudes, values, and other cognitive processes for decision-making, impulse regulation, and internal conflict resolution. Similarly, consequences for the soul that people might be able to detect seem very hard to distinguish from purely secular consequences for well-being.
However, again, the idea of souls is useful to study scientifically as a belief and social construct regardless of whether souls exist independently of culture. Plenty of research examines the relationship between religiousness and well-being in general, and some even considers the specific relationship between religiousness and the experience of meaning in life. Thanks in part to my own research efforts as an undergraduate, there's even one published study out there that reports the specific relationship between self-reported meaning in life and belief in an afterlife ($r=.32, p<.01$; Steger et al., 2010). That's the best empirical evidence of which I'm aware so far that belief in a dualistic soul relates to the subjective meaningfulness of existence.
Piedmont, R. L. (1999). Does Spirituality Represent the Sixth Factor of Personality? Spiritual Transcendence and the Five‐Factor Model. Journal of Personality, 67(6), 985–1013. http://onlinelibrary.wiley.com/doi/10.1111/1467-6494.00080/abstract.
Piedmont, R. L. (2007). Cross-cultural generalizability of the Spiritual Transcendence Scale to the Philippines: Spirituality as a human universal. Mental Health, Religion & Culture, 10(2), 89–107. Available online, URL: http://www.researchgate.net/publication/232602191_Spirituality_and_religiosity_as_cross-cultural_aspects_of_human_experience/file/32bfe50d10210133de.pdf.
Piedmont, R. L., & Leach, M. M. (2002). Cross-cultural generalizability of the Spiritual Transcendence Scale in India: Spirituality as a universal aspect of human experience. American Behavioral Scientist, 45(12), 1888–1901. http://abs.sagepub.com/content/45/12/1888.short.
Steger, M. F., Pickering, N. K., Adams, E., Burnett, J., Shin, J. Y., Dik, B. J., & Stauner, N. (2010). The quest for meaning: Religious affiliation differences in the correlates of religious quest and search for meaning in life. Psychology of Religion and Spirituality, 2(4), 206–226.