What is the origin of an arbitrary basis? Suppose I say
Let $S$ be a basis of $R^2$:
$$S = \{u_1, u_2\} = \{(1, 2), (3, 5)\} $$ 
How do I find the origin of this basis?
 A: There is no "jump" from one origin to any "new basis location".  The origin (zero vector) is the same "location" and is represented the same unique way with respect to any basis of a vector space.
In your example, the linear combination $0\cdot(1,2) + 0\cdot(3,5) = 0\cdot(1,0) + 0\cdot(0,1)$ gives the same result $(0,0)$ regardless of what basis for $\mathbb{R}^2$ is chosen.  So the coordinates of the origin with respect to any basis of a vector space are always zero coordinates, whether in two-dimensions or another vector space.

The issue of what happens to the "origin" may be better understood if we consider what happens to nonzero vectors in "translating" coordinate representations from one basis to another.
When asking "What happens to the point $(0.00001,0.00001)$?", we need to clear up any confusion about the notation for points in $\mathbb{R}^2$.  With respect to the standard basis for $\mathbb{R}^2$, these coordinates represent a point on the line $y=x$ in the Cartesian plane.
There are two computations that can be done if a new (nonstandard) basis is chosen.  Ordinarily the choice of new basis has to be described in terms the standard basis, and I think that's what we mean if we take $S = \{(1,2),(3,5)\}$ to be a different basis.  The vectors of this basis $S$ are given in terms of standard basis coordinates.
The two computations are then:
(1) How do I "translate" a point's standard coordinates into coordinates for the nonstandard basis, say $S$ as a case in point?
(2) How do I "translate" a representation in coordinates for the nonstandard basis into the standard coordinates for the same point?
Combining these two operations lets us "translate" coordinates for a point in any basis to coordinates for the same point in any other basis.  As a practical matter, if the standard coordinates for the nonstandard basis vectors are known (as they are for $S$ in particular), then computation (2) is pretty easy (just take a linear combination) and computation (1) may be slightly challenging (because it requires solving a system of linear equations to get the "new" coordinates).
