I don't know what possible connections you have in mind between the fundamental theorem of algebra and Riemann surfaces or between non-Euclidean geometries and Riemann surfaces, and I don't know why you would be particularly advocating either of those as a major impetus in the historical development. (I'm not denying the existence of connections -- all math is one, you know -- but I'm not at all seeing why those would be major actors in the story.)

If I were you, I would definitely get hold of Weyl's book. My own understanding -- not based on a study of the primary 19th century sources, but on my general experience as a mathematician -- is that Riemann was developing clear geometric intuitions for the study of algebraic curves $f(z, w) = 0$ where $f$ is a polynomial in two variables with complex coefficients. This is a huge story, related to things like elliptic functions which were much investigated during that time, and had a lot to do with the rise of modern algebraic geometry (cf. Riemann-Roch theorem, for instance, and consideration of the intrinsic geometry of curves).

I strongly advise your not asking another question on MO about this until you've had a chance to study these matters in more depth. There is generally a pretty high bar for admittance of questions.

The question has received various responses; in particular (from KConrad):

"If you consider traditional calculus to mean calculus in one variable, then that leads to complex analysis in one variable. At first it was done on C, but for more flexibility (particularly in relation to speaking about analytic continuation without awkward branch cuts) Riemann introduced the idea of doing complex analysis on a one-dimensional complex manifold, and those are essentially the same thing as Riemann surfaces"

and my subsequent response:

"[following up from you (sic) response-] Was that the primary motivation for Riemann surfaces? ... I'm trying to understand if - (and in what way if applicable) - the Riemann surface concept brought coherence to the ideas in analysis{star}, geometry, and algebra (specifically, the Fundamental Theorem of Algebra (as this theorem, as far I'm aware, required introduction of complex analysis / complex analytic ideas to prove [at least as one of its earlier / earliest proofs])). [{star} to refer to the above statement here: "[theory of Riemann surfaces] is [the] culmination of much of traditional calculus"]".

I'm not searching for textbooks or technical material [except books on the historical genesis and motivating reason(s) for Riemann surfaces]. ... More specifically, [as above], the relation between the birth of Riemann surface theory in light of developments within the three primary branches: geometry, analysis, and algebra.

(Within geometry: the emergence of non-Euclidean geometries: (especially as most of non-Euclidean geometries use the notion of 'manifold' - which, if I'm not mistaken - came from the context of studying Riemann surfaces);

Within analysis: -- KConrad's statement above -- ;

and

Within algebra: the fact that a good number of proofs of the Fundamental Theorem of Algebra require the use of ideas from complex analysis (and, indirectly, relate to Riemann surfaces).)

I want to get an expert's take on the thread's outlined above, and the relation between each other and the concept of Riemann surfaces.

So, basically, that is my question.

Any feedback would be greatly appreciated.