You asked many questions, here are a few things related to some of them:
(1) What does $H^i(X,\mathcal O_X)$ mean?
In the examples you quoted it already shows that it depends a lot on $X$ what these groups mean.
(a1) if $X$ is smooth and projective, then there is the Hodge decomposition, which I am sure is something every topologists appreciate. It tells you that the singular cohomology groups $H^m(X,\mathbb C)$ may be decomposed as the direct sum of the Dolbeault cohmology groups $H^{p,q}(X)$ for $p+q=m$. One of these is isomorphic to $H^m(X,\mathcal O_X)$, so one possible answer to your question is that these cohomology groups give you a piece of the singular cohomology. A particular interesting case is $m=1$. Then $b_1=h^{0,1}+h^{1,0}=2\cdot h^{0,1}$. So, the vanishing or non-vanishing of $H^1(X,\mathcal O_X)$ is equivalent to the same for $H^1(X,\mathbb C)$.
(a2) still in the projective case, there is a duality, called Serre duality, between cohomology groups of $\mathcal O_X$ and those of $\omega_X$, the sheaf of top differential forms (i.e., the determinant of the cotangent bundle). So, $\dim H^i(X,\mathcal O_X)=\dim H^{n-i}(X,\omega_X)$ where $n=\dim X$.
(b) If $X$ is the complement of a closed subset in an affine variety, then higher cohomology of any coherent sheaf is isomorphic to a shifted local cohomology (with supports in the complement). In other words, $H^i(X,\mathcal O_X)\simeq H^{i+1}_Z(\bar X, \mathcal O_{\bar X})$ for $i>0$ where $X=\bar X\setminus Z$. Local cohomology tends to be big (if not zero), and that's the reason for that example you mention.
Of course, now you can ask what local cohomology means, but I'll leave that for another answer/question.
(2) Replacing regular with analytic functions.
Quotients of analytic functions are analytic on their domains of definitions, so while you could define regular analytic functions as those that are locally quotients of analytic functions, you would not actually change the category. The point is that in AG, polynomials are the functions that we can originally define, but it makes sense that as long as the reciprocal of a function exists, then we should be able to use that reciprocal as a regular function. However, those are no longer polynomials, so we need this sort of extended definition.
Best Answer
One example is given by Enriques surfaces in characteristic $2$. There are three types depending on the value of $\mathrm{Pic}^\tau$ (as a group scheme) which can be either $\mathbb Z/2$, $\mu_2$ or $\alpha_2$. In the first case $\omega_X$ is the generator so in particular it is non-trivial and $H^2(X,\mathcal O_X)=0$ (using of course Serre duality). In the two other cases $\omega_X$ is trivial (as it is numerically trivial and all numerically trivial line bundles are trivial) so that $h^2(X,\mathcal O_X)=1$. Now, $\alpha_2$ can be deformed to both $\mathbb Z/2$ and $\mu_2$ and such deformations can be lifted to deformations of Enriques surfaces (in fact $\mathrm{Pic}^\tau$ is flat in families of Enriques surfaces and the functor from deformations of the surfaces to those of $\mathrm{Pic}^\tau$ is formally smooth - Liedtke: arXiv:1007.0787, Ekedahl-Shepherd-Barron: unpublished). If we pick a connected family of Enriques surfaces with some special value being an $\alpha_2$-surface and generically $\mathbb Z/2$, then we get an example.
Such an example can be constructed (very) explicitly without deformation theory. Here is a semi-explicit construction which works in any positive characteristic. Fixa a a group scheme $A$ of order $p$ over $\mathbb A^1$ localised at $0$ (say) which is $\alpha_p$ at $0$ and $\mathbb Z/p$ elsewhere. By the Godeaux construction (which Raynaud - Prop. 4.2.3, p-torsion du schema de Picard, Astérisque 64 - showed works for such families) there is a free action of $A$ on a flat complete intersection $Y$ (of any dimension, which we assume is $\ge 2$) such that $X=Y/A$ is smooth (note that contrary to the case of an étale group scheme $Y$ will not be smooth). As $Y$ is a complete intersection we have that $\mathrm{Pic}^\tau_Y=0$ and from that it follows that $\mathrm{Pic}^\tau_X=A$. Now, $H^1(-,\mathcal O_-)$ is the tangent space of $\mathrm{Pic}^\tau$ so it is zero outside of $0$ and $1$-dimensional at $0$. (By being careful one can get Enriques surfaces for $p=2$, this I guess was the inspiration for the Godeaux construction).