First a warning: I don't know much about either algebraic topology or its uses of physics but I know of some places so hopefully you'll find this useful.
Topological defects in space
The standard (but very nice) example is Aharonov-Bohm effect which considers a solenoid and a charged particle. Idealizing the situation let the solenoid be infinite so that you'll obtain ${\mathbb R}^3$ with a line removed.
Because the particle is charged it transforms under the $U(1)$ gauge theory. More precisely, its phase will be parallel-transported along its path. If the path encloses the solenoid then the phase will be nontrivial whereas if it doesn't enclose it, the phase will be zero. This is because $$\phi \propto \oint_{\partial S} {\mathbf A} \cdot d{\mathbf x} = \int_S \nabla \times {\mathbf A} \cdot d{\mathbf S} = \int_S {\mathbf B}\cdot d{\mathbf S}$$ and note that $\mathbf B$ vanishes outside the solenoid.
The punchline is that because of the above argument the phase factor is a topological invariant for paths that go between some two fixed points. So this will produce an interference between topologically distinguishable paths (which might have a different phase factor).
Instantons
One place where homotopy pops up are Instantons in gauge theories.
Specifically, if you consider a Yang-mills theory in ${\mathbb R}^4$ (so this means Euclidean time) and you want the solution (which is a connection) to have a finite energy then its curvature has to vanish at infinity. This allows you to restrict your attention to $S^3$ (this is where the term instanton comes from; it is localized) and this is where homotopy enters to tell you about topologically inequivalent ways the field can wrap around $S^3$. Things like these are really big in modern physics (both QCD and string theory) because instantons give you a way to talk about non-perturbative phenomena in QFT. But I am afraid I can't really tell you anything more than this. (I hope I'll get to study these things more myself).
TQFT
Last point (which I know nearly nothing about) concerns Topological Quantum Field Theory like Chern-Simons theory. These again arise in string theory (as does all of modern mathematics). And again, I am sorry I cannot tell you more than this yet.
I've been thinking about divergent series on and off, so maybe I could chip in.
Consider a sequence of numbers (in an arbitrary field, e.g. real numbers) $\{a_n\}$. You may ask about the sum of terms of this sequence, i. e. $\sum a_n$. If the limit $\lim_{N\rightarrow\infty} \sum^N |a_n|$ exists then the series is absolutely convergent and you may talk about the sum $\sum a_n$. In case the limit does not exist but $\lim_{N\rightarrow\infty} \sum^N a_n$ exists then the sequence is conditionally convergent, and as (I assume) Carl Witthoft commented above there is a theorem stating that you may sum the sequence in a different order and get a different result for the limit. In fact by judiciously rearranging you may get any number desired. I included this just to mention that although divergent series may seem most bizarre, in the sense of summing terms and that by each term it gets nearer a limit, only the absolutely convergent series make connection with our intuiton. So we may ask about making sense of series in general.
As G. H. Hardy's "Divergent Series" explain in page 6, the trick is to understand that our usual notion of sum of a serie is a way to define something we call a "sum". In other words given a sequence we have a map that atributes to this sequence a number. The "sum map" being the trivial operation of summing the terms if the series is absolutely convergent. The idea behind divergent series is to realize that this map althogh in a sense canonical is not unique.
To be more specific, consider the space $V$ of all sequences together with operations of addition and scalar multiplication (given two sequences $\{a_n\}$ and $\{b_n\}$ and a number $\lambda$ we define addition by $\{a_n\}+\{b_n\}=\{a_n+b_n\}$ and scalar multiplication by $\lambda\cdot\{a_n\}=\{\lambda a_n\}$). Now the space of sequences with these operations is a (infinite-dimensional) vector space (there is a good question about coordinates, since I am assuming one specific basis here, but let's not worry about this now). The absolutely convergent series can be seen to form a subspace $U$ of $V$ and the "sum" $S$ is just a linear functional on this subspace, $S:U\rightarrow\mathbb{R}$. The problem is that this functional is not defined anywhere else.
So to make sense of divergent series one asks if is there another map $S'$ defined on a subspace $W$ that contains $U\subset W$ such that when restricted to this subspace is just the usual sum, i.e. is there $S'$ such that $S'|_U=S$?
And in fact many such functionals do exist. And each of these we call a different "summation method" in the sense that it attributes a value to a sequence and that when such sequence corresponds to a convergent series it gives the usual values.
For instance, Cesaro Summation says that maybe the series does not converge because it keeps oscillating (like the $1-1+1\cdots$ you mentioned). Then we could take the arithmetic mean of the partial sums $s_n=a_1+\cdots a_n$ and define the Cesaro sum of the sequence $\{a_n\}$ as $\lim_{N\rightarrow\infty}\frac{1}{N}\sum^N s_n$. Is not hard to see that this gives the usual result for convergent series (although a bit obscure that it is a linear map), but it also gives $1/2$ to the alternating series of $1-1+1\cdots$. So one must give a new meaning to the word "sum", and then you can get new results. For instance Fejer's theorem roughly states that (given mild conditions) the Fourier series of a function may not be stricty convergent, but it is always summable in the sense of Cesaro. So it tells you that the worst divergence that appears in Fourier series is of oscillating type, i. e. the series never diverges to $\pm\infty$. Furthermore by Cesaro summing you can tell around which value the series oscillates about. But this does not "sum" the series in the sense of making it convergent in the usual sense.
Other ideas for functionals is by analytic continuation. The most obvious is the geometric series $\sum x^n=\frac{1}{1-x}$. Is is only convergent in the $|x|\leq 1$ radius, but one may use analytic continuation to turn around the problem at $x=1$ and say that $1+2+4+\cdots=\frac{1}{1-2}=-1$. In this case is easier to picture the linear algebra idea. The space $U$ is of all geometric series with $|x|\leq 1$ and $S$ is the usual sum. Now we introduce a functional $S'$ by $S':W\rightarrow\mathbb{R},x\mapsto\frac{1}{1-x}$ so that $W$ is now every geometric series except the one with $x=1$. A functional which reduces to this case for the geometric series but does it also for other power series is Abelian Summation.
So the idea is not to "sum" the series really (only absolutely convergent series sum in usual sense) but to redefine the notion of sum by generalizing the concept and then using this different notion to attribute a finite value to the serie through the corresponding sequence. This finite value should tell you something about the series, like the Cesaro sum tells you around which value the series oscillates or like Abel sum is able to reconstruct the function that generated the series. So summation methods are able to extract information from divergent series, and this is how to make sense of them.
With respect to the physics, is important to stress that perturbative series in quantum field theory are (generally) divergent but neither renormalization nor regularization have to do (fundamentally) with "summing" divergent series (zeta regularization being one technique, not mandatory, although useful). Rather what does occur is that it sometimes one gets a asymptotic serie in perturbation theory. In general there are different functions with the same asymptotic serie, but with supplementary information it may, or may not, occur that one can uniquely find the function with that specific asymptotic series. In this case one can use a summation method known as Borel Summation to fully reconstruct the entire function. When such thing happens in QFT is normally associated with the presence of some sort of instanton. You can take a further look in S. Weinbergs "Quantum Field Theory Vol. 2", page 283. So the idea is to get non-perturbative information out of the perturbation series, and not to tame some sort of infinity. Renormalization is something completely different (and much worse since in fact it is highly non-linear for starters).
For further information try finding a copy of Hardy's book (it's a gem), or for the linear algebra babble J. Boos, F. P. Cass "Classical and Modern Methods In Summability".
Best Answer
In QFT and String theory this divergences usually shows up when we take a continuum limit. As we have learned in the Wilson's Renormalization Group, the continuum limit in Statistical Mechanics and Quantum Mechanics is very subtle. The divergences shows up if we are naive about this limit.
There are various ways to proceed with this continuum limit, almost all characterized by the following steps:
Symmetries are very important here. When we do the first step, the regularization, there are two possibilities: preserve the symmetry or not. If we are able to preserve the symmetry, things are more straightforward. If we don't, we need to impose the symmetry by hand through the calculation and check by hand if this is really possible, i.e. if there is no anomaly.
In your particular case, you want to calculate: $$ S=\sum_{n=0}^{\infty}n $$ the obvious regularization will be a hard cut-off: $$ S(N)=\sum_{n=0}^{N}n $$ and then at the end of the calculation we make $N\rightarrow\infty$. This is obvious divergent. Turns out that this regularization does not preserve the symmetries of the problem. In order to preserve those symmetries you should add counter-terms to this quantity: $$ S(N)=\sum_{n=0}^{N}n + S_{ct}(N) $$ such that the symmetry is restored. This counter-terms should be compatible with the renormalization conditions of step 2.
There are others regularization like the Heat Kernel Regularization: $$ S(\varepsilon)=\sum_{n=0}^{\infty}ne^{-n\varepsilon} $$ this still does not preserve the symmetries of the problem but is more easy to impose the symmetries through the calculation and identify the counter-terms.
There is a regularization that in almost all the cases preserve all the symmetries of the problem, the Zeta regularization: $$ S=\zeta (-1) $$ and so, there is no need for counter-terms.
There is a very nice blog post about it here