This question has two main parts:
1. Spherical coordinates and their basis vectors
Dealing with spherical coordinate system, it is quite clear its introduction, one picks two mutually perpendicular directions which enables to introduces coordinates $(r,\theta,\phi)$.
To get the basis, one can pick e.g. a covariant basis as tangents to the coordinate lines and it happens that the direction of the basis vectors changes. In Cartesian system, any vector in $\mathbb{R}^3$ can be written as a linear combination of $\vec{e}_x,\vec{e}_y,\vec{e}_z$, and it is very clear what that means geometrically. This basis generates the whole $\mathbb{R}^3$ volume, and I really need all the three components of the basis.
On the other hand, any point in $\mathbb{R}^3$ is just $r \vec{\hat{e}}_r$, $\vec{\hat{e}}_r$ being the normalized vector tangent to the radial coordinate and it seems that I need only one basis vector (although I still need 3 different coordinates). At the same time, one usually introduces the line element $d\vec{r} = dr\,\vec{\hat{e}}_r + d\theta\, r\vec{\hat{e}}_\theta + d\phi\, r\sin{\theta}\, \vec{\hat{e}}_\phi$ which clearly should not be parallel to any $\vec{\hat{e}}_r$.
I think this has something to do with the idea introduced in the Cartesian system that e.g. for adding vectors, one just moves everything into the origin and does the operation there and the vectors always seem originate in the origin and in general, all vectors that have the starting point anywhere in space are considered the same as if one shifts them to the origin. However, in curvilinear systems this probably cannot be done. But this is just my guess as I have never read anything like this.
The core of this question is, what does it then mean a linear combination of $\vec{e}_r,\vec{e}_\theta,\vec{e}_\phi$? Additionally, do these three vectors form a basis of a vector space and if so, what vector space? Should I thing of $\mathrm{Span}\{\vec{e}_r,\vec{e}_\theta,\vec{e}_\phi\}_{(r_0, \theta_0,\phi_0)}$ to be completely different from $\mathrm{Span}\{\vec{e}_r,\vec{e}_\theta,\vec{e}_\phi\}_{(r_1,\theta_1,\phi_1)}$ and should I consider each a Cartesian space and them to be related by rotation and translation?
Because if they are just local Cartesian systems, then what is the point of lamé coefficients? This closely relates to the second question.
2. Cross product in spherical coordinates
After introducing covariant derivative, I decided to derive all the usual nabla expressions (gradient, divergence, rot, laplace) in spherical coordinates from Christofel symbols. As a part of that, the book I am reading (Introduction to tensor analysis and calculus on moving surfaces by Grinfeld), introduces the Levi-Civita tensor ($\varepsilon^{ijk}$) related to the permutation Levi-Civita symbol ($e^{ijk}$) through the determinant of the covariant metric tensor as: $$ \varepsilon^{ijk}= \sqrt{\det(Z_{..})}^{-1} e^{ijk}\\ \varepsilon_{ijk}= \sqrt{\det(Z_{..})}\, e_{ijk} $$
Then I was curious to calculate cross product in spherical coordinates. Through the properties of $e^{ijk}$, i could show that for $\vec{w} = \vec{u}\times\vec{v} = \varepsilon^{ijk} u_j v_k \vec{Z}_i$, where $\vec{Z}_i$ is just the notation for ith covariant basis vector:
$$ \vec{w} = \frac{1}{r^2 \sin{\theta}}\det \begin{pmatrix} \vec{Z}_1 & \vec{Z}_2 & \vec{Z}_3\\ u_1 & u_2 & u_3 \\ v_1 & v_2 & v_3 \\ \end{pmatrix} $$
where the covariant basis for spherical coordinates in terms of cartesian coordinates should be: \begin{align} \vec{Z}_1 &= \phantom{-r}\sin{\theta}\cos{\phi}\,\vec{e}_1 + \phantom{r}\sin{\theta}\sin{\phi}\,\vec{e}_2 + \phantom{r}\cos{\theta}\,\vec{e}_3 \\ \vec{Z}_2 &= \phantom{-}r\cos{\theta}\cos{\phi}\,\vec{e}_1 + r\cos{\theta}\sin{\phi}\,\vec{e}_2 - r\sin{\theta}\,\vec{e}_3 \\ \vec{Z}_3 &= -r\sin{\theta}\sin{\phi}\,\vec{e}_1 + r\sin{\theta}\cos{\phi}\,\vec{e}_2 \end{align}
The determinant will spit some linear combination of the covariant basis $\{\vec{Z}_1, \vec{Z}_2, \vec{Z}_3\}$ and as I asked above, i have not much of an idea what that really means. The second question here is whether the cross product is correct (calculated as the determinant).