How do the unit vectors in spherical coordinates combine to result in a generic vector?

The curvilinear unit vectors are tricky in that their expression depends on which point the vector corresponds to. For example, the vector $\mathbf v=v_x\,\hat x$ can always be expressed in this way no matter where the vector "is located". However, if this vector $\mathbf v$ is located on the x-axis, then it only has a $\hat r$ component using spherical unit vectors. If $\mathbf v$ is located on the $-y$ axis, then it only has a $\hat\phi$ component using spherical unit vectors. Conversely, this means that saying a vector is, for example, $\mathbf v=v_r\,\hat r$ is not enough to determine the actual direction of the vector (we just know it is pointing away from or towards the origin, but not from where it is pointing). This is typically why it is recommended to convert to Cartesian unit vectors before performing vector integrals in general, as the Cartesian unit vectors do not have this dependence.

In general, if you have some vector $\mathbf v=v_x\,\hat x+v_y\,\hat y+v_z\,\hat z$ located at the spatial point $(x,y,z)$, then we can transform the representation using the following transformations: $$\hat x=\sin\theta\cos\phi\,\hat r+\cos\theta\cos\phi\,\hat\theta-\sin\phi\,\hat\phi$$ $$\hat y=\sin\theta\sin\phi\,\hat r+\cos\theta\sin\phi\,\hat\theta+\cos\phi\,\hat\phi$$ $$\hat z=\cos\theta\,\hat r-\sin\theta\,\hat\phi$$

where $$\theta=\tan^{-1}\left(\frac{\sqrt{x^2+y^2}}{z}\right)$$ $$\phi=\tan^{-1}\left(\frac{y}{x}\right)$$

Do we actually combine all 3 unit vectors in spherical coordinates to obtain a certain vector...

Yes, you just need to also specify where the vector is. In other words, saying $\mathbf v=v_r\,\hat r+v_\theta\,\hat\theta+v_\phi\,\hat\phi$ is not sufficient to specify the vector.

Conceptionally, there's a difference between points and vectors. Given a set of coordinates, each point carries around its own frame induced by the coordinate lines, a basis of the vector space of tangent vectors rooted at that point.

In flat Euclidean space, any such frame extends globally, and we can even use it to describe points in terms of position vectors relative to some chosen point of origin. In that case, we could just fix one frame to describe everything, but even when that's possible, it can still make sense to instead phrase things in terms of the different frames that change from point to point. There's value in, say, having an 'up' direction that always points away from the center of the earth, no matter where you are.

You're right to state that in spherical coordinates, you can represent a vector as just $r\mathbf{\hat{e}_{r}}$.

The angular quantities come in when you are either attempting to understand what $\mathbf{\hat{e}_{r}}$ is in terms of another basis, say a Cartesian one—there the angles show up as you'd expect—or if you're calculating rates of change of a vector in spherical coordinates, in which case the derivatives of $\mathbf{\hat{e}_{r}}$ will have components in the $\mathbf{\hat{e}_{\theta}}$ and $\mathbf{\hat{e}_{\phi}}$ directions.