Yes. The category of algebraic spaces is the smallest subcategory of the category of sheaves of sets on Aff, the opposite of the category of rings, under the etale topology which (1) contains Aff, (2) is closed under formation of quotients by etale equivalence relations, and (3) is closed under disjoint unions (indexed by arbitrary sets). An abstract context for such things is written down in "Algebraization of complex analytic varieties and derived categories" by Toen and Vaquie, which is available on the archive. Toen also has notes from a "master course" on stacks on his web page with more information. It might be worth pointing out that their construction of this category also goes by a two-step procedure, although in their case it's a single construction performed iteratively (and which stabilizes after two steps). This is unlike the approach using scheme theory in the literal sense, as locally ringed topological spaces, where the two steps are completely different. After the first step in T-V, you get algebraic spaces with affine diagonal. Also worth pointing out is that their approach is completely sheaf theoretic. The only input you need is a category of local models, a Grothendieck topology, and a class of equivalence relations. You then get algebraic spaces from the triple (Aff, etale, etale). But the general machine (which incidentally I believe is not in its final form) has nothing to do with commutative rings. I think it would be interesting to plug opposites of other algebraic categories into it.
As far as I know, the prototypes of obstruction theories in algebraic geometry originated from the more general Kodaira-Spencer theory of deformation of complex manifolds [see Kodaira-Spencer, On deformations of complex analytic structures I-II-III, Annals of Math., 1958-1960].
The crucial question was
When an "infinitesimal" deformation of a compact complex manifold $M$ (in particular, of a complex projective variety) gives rise to a "genuine" deformation of $M$, i.e. a deformation over a disk?
The answer to this question is contained in the following result, see [Kodaira, Complex manifolds and deformations of complex structures, Theorem 5.1]:
Theorem. Suppose given a compact complex manifolds $M$, and $\theta \in H^1(M, \Theta_M)$. In order that there may exist a complex analytic family $\omega \colon \mathcal{M} \to B$ such that $\omega^{-1}(0)=M$ and $(dM_t/dt)_{t=0}=\theta$, it is necessary that $[\theta, \theta]=0$ holds.
In fact, Kodaira explicitly says that "if $[\theta, \theta] \neq 0$ there is no deformation $M_t$ with $\omega^{-1}(0)=M$ and $(dM_t/dt)_{t=0}=\theta$. In this sense, we call $[\theta, \theta] \in H^2(M, \Theta_M)$ the obstruction to the deformation of $M$".
Of course Kodaira was well aware that the condition $[\theta, \theta]=0$ is not sufficient in general, since there can be higher-order obstructions, corresponding to the need of finding higher and higher truncations of the solution of the Maurer-Cartan equation governing the deformation of the given complex structure. Only if all these obstructions vanish we can hope to find our complex analytic family $\mathcal{M}$.
In Kodaira's words, "Thus we have infinitely many obstructions to the deformations of $M$. In view of this fact we call $[\theta, \theta]$ the primary obstruction".
The obstruction theories coming later in algebraic geometry, as far as I know, were build up in order to rephrase and extend Kodaira-Spencer theory in a completely algebro-geometrical setting (for instance, making possible deformation theory in characteristic $p$), in order to deform objects different from complex structures, such as coherent sheaves, subvarieties, maps, and in order to understand the difference between the deformations in the analytic sense and those in the algebraic sense ("algebraization problem").
Best Answer
If you take, say, set of real points of the group-scheme $O(n)$, i.e., $O(n, {\mathbb R})$, then you recover the usual orthogonal (real Lie) group, which you know as $O(n)$. Same applies to $SL(n)$, etc. There is one case when this does not work well, namely when you deal with character varieties. For instance, take $\pi$, say, the free group on two generators, and try to form the quotient $Q=Hom(\pi, SU(2))/SU(2)$. The standard way to do this is to consider the corresponding character variety (or, rather, affine scheme) $X$ and take its set of real points. However, the result will contain both equivalence classes of representations of $\pi$ to $SU(2)$ (as you expected), but also equivalence classes of representations to $SL(2, {\mathbb R})$! The easiest way to see this is to realize that the coordinate ring of $X$ is generated by traces of the elements $A, B, C=AB$ of $\pi$ (where $A, B$ are the free generators). To get the set of real points, you need to use points with real traces, so you end up with the elements of both real Lie groups $SU(2)$ and $SL(2, {\mathbb R})$. This is rather annoying, but one can learn to live with this problem. Namely, in order to isolate $Q=Hom(\pi, SU(2))/SU(2)$ inside $X({\mathbb R})$, you impose also some inequalities, so $Q$ becomes a real semi-algebraic subset. Same problem appears if you consider $Hom(\pi, SL(2, {\mathbb R}))$: Character variety will give you unitary representations as well. The standard way to deal with this problem (in Teichmuller theory) is to consider not all representations to $SL(2, {\mathbb R})$, but only discrete and faithful ones, so that the commutator $[A,B]$ maps to elements of the fixed trace. Then you can form the (topological) quotient by $SL(2, {\mathbb R})$ by taking slice, i.e., restricting to representations $\rho$ so that the (attractive, repulsive) fixed points of $\rho(A)$ are $0, \infty$ and the attractive fixed point of $\rho(B)$ is $1$.
Addendum: More generally, in all "interesting" case I know, the desired quotient can be constructed without algebraic geometry. Suppose you are interested in $R:=Hom(\pi, O(n,1))$ and the group $\pi$ is "nonelementary", i.e., is not virtually abelian. Then, in $R$ consider the open subset $R'$ consisting of representations $\rho$ so that $\rho(\pi)$ does not fix a point in $S^{n-1}$. For instance, $R'$ will contain all discrete and faithful representations. Then you can just take the "naive" quotient $Q=R'/O(n,1)$ instead of the (set of real points of) character variety $X({\mathbb R})$. Then $Q$ will embed in $X({\mathbb R})$.
Concerning existence of a slice: The same argument I described works for representations to $SL(2, {\mathbb C})$. However, if you consider representations to $O(n,1), n\ge 4$, there is no (in general) global slice. However, locally, it does exist, see e.g. Slice Theorem for the general information about slices for group actions.
Recommended reading: D. Johnson and J. Millson, Deformation spaces, associated to compact hyperbolic manifolds, in "Discrete Groups in Geometry and Analysis" (Papers in honor of G. D. Mostow on his sixtieth birthday), 1984. I read this paper as a graduate student and still find it useful.
By the way, here is where algebraic viewpoint is definitely superior to the Lie theoretic. Suppose that you are interested in understanding local structure of the analytic variety $Hom(\pi, G)$, where $G$ is a real Lie group, i.e., what singularity it has at a representation $\rho: \pi\to G$. In general it is a rather difficult problem as singularities could be "arbitrarily complicated." See M.Kapovich, J.Millson, "On representation varieties of Artin groups, projective arrangements and the fundamental groups of smooth complex-algebraic varieties", Math. Publications of IHES, vol. 88 (1999), p. 5-95, one of the main results there is that singularities of character varieties could be arbitrary (defined over ${\mathbb Q}$).
So, assume that $G=\underline{G}({\mathbb R})$, where $\underline{G}$ is an algebraic group (scheme) over reals. Let $A$ be an Artin local ${\mathbb R}$-ring with the projection to the residue field $\nu: A\to {\mathbb R}$. Then the set of $A$-points $G_A:=\underline{G}(A)$ is a certain nilponent extension of the Lie group $G$ with the quotient $\nu_G: G_A\to G$ induced by $\nu$. Then, instead of analyzing the scheme $Hom(\pi, \underline{G})$ at $\rho$, you consider the collection of real-algebraic sets $Hom_{\rho}(\pi, G_A)\cong Hom_{\rho}(\pi, \underline{G})(A)$, consisting or representations $\tilde\rho: \pi\to G_A$ which project to $\rho$ under $\nu_G$. The point is that the collection of real-algebraic sets $Hom_{\rho}(\pi, G_A)$ "knows everything" (and even more!) about the singularity of $Hom(\pi, G)$. For instance, to recover the (Zariski) tangent space $T_\rho Hom(\pi, G)$, you just take $A$ to be the "dual numbers", which is the quotient ${\mathbb R}[t]/(t^2)$. Then $$ T_\rho Hom(\pi, G)\cong Hom_{\rho}(\pi, G_A)$$ for this choice of $A$.
This staff is explained in the paper W.Goldman, J.Millson, The Deformation Theory of Representations of Fundamental Groups of Compact Kahler Manifolds, Publ. Math. I.H.E.S.; 67 (1988).