How are the LCAO ansatz and Slater determinant ansatz for the wavefunction connected?
I found the question somewhat confusing. I think it is because mixing up some concepts.


  I understand that for the basis set expansion in an orthonormal basis,
  one can obtain the coefficients by the inner product  $$
 \left&lt;\phi_i|\Psi \right&gt;=c_i $$


Please, think about what you mean by the coefficients and where the functions live, before continue reading. 

As you thought this expression as an inner product (which is right), both functions must live in the same space so they have the same number of coordinates. What you are doing with that inner product is projecting $\Psi$ over the $\phi_i$. If $\{\phi_i\}$ is an orthogonal set, then you can make the expansion you wrote. But, if the are not, then such expansion is meaning less. You still can make expansions that recover $\Psi$  (in fact, an infinite number of them). So, I get confused about which coefficients are you referring to. As for example, $\phi_1$, appears in both: many Slater's det. and in many terms in one (or more) Slater's det., to ask  which is its coefficient in the Slater det. makes not sense, except if you want to expand the Slater det. into a LCAO which is not possible (but you still can project it).

See below.


  But how does this work if the wavefunction is multidimensional like the slater determinant wavefunction ? 


First, note that even for 1-electron problems the problem is not mono-dimensional. So, the case of Slater's determinants is exactly the same. You only need to integrate over every coordinate.


  To my knowledge only basis functions of one coordinate are used in
  calculations, like 3D Gaussians.


3D-Gaussians have three coordinates. It is obvious using Cartesian coordinates. Under spherical coordinates they only depends on $r$, but they still are functions of $\theta$ and $\phi$. Although you can take only one coordinate ($r$) for its evaluation, it is not true for computing an arbitrary inner product.


  Assuming that i have a wavefunction given in slater determinant form,
  how would i find the coefficients of the basis functions ?


At first, take in mind that those coefficients are for expanding the Slater det. into the subspace spanned by your $\{\phi_i\}$. That is, it they are not an complete set, you only will get an approximation. You just have to perform those integrations. In your example, the $c_i$ will not be a function of $x_2$, because you have to integrate over every coordinate.

I am not sure if I understand correctly your doubts. Please, do not hesitate in ask for clarification. Maybe this answer could be useful.