What strikes me about the first fifty years of homology theory (from Poincaré to Eilenberg-Steenrod's book) is that the development was as much about stripping away unnecessary complication as about increasing sophistication. A famous example is singular homology, which was found very late, by Eilenberg. The construction as we know it presumably seemed too naive to Lefschetz, who misguidedly devised a theory of oriented simplices, and inadequate to those who were interested in general (not locally path connected) metric spaces.
I want to suggest that this process of stripping away is relevant to the introduction of cohomology and its product. (Cf. Dieudonné's "History of algebraic and differential topology", pp.78-81). I won't directly answer the questions, but will suggest that one of the motivations for cohomology came from an application of Pontryagin duality which was rendered obsolete by the new theory.
Alexander wrote up his Moscow conference talk, with improvements suggested by Cech, in a 1936 Annals paper (vol. 37 no. 3) (JSTOR link). In it, he proposes the cohomology ring ("connectivity ring") as a fundamental homological invariant of a space. In the introduction he hints at the line of thought that led him to the cohomology ring. The relation between cycles and differential forms is mentioned (without citation of de Rham), but what looks more surprising to modern eyes is the comment that the theory of cycles "has been very greatly perfected by Pontrjagin's cycles with real coefficients reduced modulo 1".
Pontryagin had recently developed his duality theory for locally compact abelian groups (Annals, 1934) in order to apply it to Alexander duality (again, Annals, 1934). If $K$ is a compact polyhedral complex in $\mathbb{R}^n$, there is a linking form which gives a pairing between $k$-cycles of $K$ and $(n-k-1)$-cycles of $\mathbb{R}^n-K$ and, in modern terms, induces an isomorphism of $H_k(K)$ with $H^{n-k-1}(\mathbb{R}^n-K)$. Alexander's formulation equated the Betti numbers over a field (mod 2, initially - Dieudonné p. 57) of $K$ and its complement, but it was understood that the full homology groups of $K$ and $\mathbb{R}^n-K$ need not be isomorphic. Pontryagin showed that if one takes a Pontryagin-dual pair of metric abelian groups, say $\mathbb{Z}$ and $\mathbb{T}$, so that each is the character group of the other, then $H_k(K;\mathbb{T})$ is Pontryagin-dual to $H_{n-k-1}(\mathbb{R}^n-K;\mathbb{Z})$ via the linking form.
From Alexander's introduction:
Now, if we use Pontrjagin's cycles, the $k$th connectivity [homology] group of a compact, metric space becomes a compact, metric group. Moreover, by a theorem of Pontrjagin, every such group may be identified with the character group of a countable, discrete group. This immediately suggests the advisability of regarding the discrete group, rather than its equivalent (though more complicated) metric character group as the $k$th invariant of a space.... One decided advantage of taking the discrete groups...as the fundamental connectivity groups of a space is that we can then take the product...of two elements of the same or different groups.
Guided by Pontryagin's generalisation of his own duality theorem, Alexander finds a simple construction that supersedes Pontryagin's as a basic invariant. (The universal coefficient theorem gives a modern perspective on why Pontryagin's choice of coefficient groups works. I must admit, his formulation of duality is very clean.)
Can anyone comment on Kolmogorov's route to cohomology?
ADDED. On obstruction theory: Charles Matthews's comments draw attention to a 1940 paper of Eilenberg. The MathSciNet review of that paper (by Hurewicz, whose homotopy groups, useful for obstruction theory, date from 1935-36) points me to its 1937 forerunner by Whitney, "The maps of an $n$-complex into an $n$-sphere" (Duke M.J. 3 (no.1), 51-55). This work, too, was presented at the Moscow conference in 1935. Though the topic is different, Whitney's introduction closely resembles Alexander's:
The classes of maps of an $n$-complex into an $n$-sphere were classified by H. Hopf in 1932.
Recently, Hurewicz [1935-6] has extended this theorem by replacing the sphere with more general spaces. Freudenthal [1935] and Steenrod have noted that the theorem and proof are simplified by using real numbers reduced mod 1 in place of integers as coefficients in the chains considered. We shall give here a statement of the theorem that seems most natural; the proof is quite simple.... The fundamental tool of the paper is the notion of "coboundary"; it has come into prominence in the last few years.
Best Answer
Sheaf cohomology is a powerful tool, but it isn't a replacement for all of basic algebraic topology. For example, fundamental groups and homology some topics that would get lost. And these topics are certainly relevant to algebraic geometry. Also, as pointed out in the comments, you would lose valuable intuition if you just stuck to the sheaf cohomology viewpoint.
Let me expand my original answer a bit. Let me focus on the simplest example, where $X$ is a smooth complex projective curve of genus $g$. One learns in topology that $X$ is obtained by identifying the sides of a $2g$-gon in the standard way. One can use this to extract 2 things about $X$.
One gets the homology $H_1(X,\mathbb{Z})=\mathbb{Z}^{2g}$, with its intersection pairing equal to the standard symplectic form. For an algebraic geometer, this corresponds to the lattice of the Jacobian of $X$ together with its Riemann form. In particular, this is a principal polarization.
Also one gets the familiar presentation of the fundamental group $$\pi_1(X)= \langle a_1\ldots a_{2g}\mid [a_1,a_2]\ldots[a_{2g-1}, a_{2g}]\rangle$$ But why should an algebraic geometer care about this? Answer: because it tells us what etale covers of $X$ look like. The etale fundamental group of $X$ is the profinite completion of the above group. This is also true for the prime to $p$ part if $X$ lives in positive characteristic by lifting. Note that Grothendieck uses this reduction to the topological case in SGA1.