Use of basis set in DFT (Density Functional Theory)
This answer only deals with the most common variety of Density Functional Theory, namely, Kohn-Sham DFT. This is what most people mean by "DFT", but, as noted in the comments, things such as orbital-free DFT exist.

Kohn-Sham DFT was created to solve a historical problem of DFT: The electronic kinetic energy term (as used in the Thomas-Fermi model) was not accurate enough. Kohn and Sham proposed to make use of what is typically called a "fictitious, non-interacting reference", that is, a wave-function whose main purpose would be to yield a density to be used in all other (nucleus-electron attraction, Coulomb, exchange, and correlation energy/potential) and whose secondary function is to provide the kinetic energy, which is accurate from wave-function theory. This wave-function works just like a Slater determinant and is called the Kohn-Sham determinant. See this question for pointers on where to read more: Equivalent of Szabo and Ostlund book for DFT.

As often in DFT, practicality showed a way that was justified later. When implementing this idea, one learns that one can basically use a HF code and replace the exchange term by the exchange-correlation (XC) potential (which is projected back on to the AO basis set). The potential needs to evaluated numerically on a grid, a procedure that yields the XC energy for almost all choices of density functionals. Thus one obtains the energy and the KS operator (the equivalent of the Fock operator) and can perform a SCF procedure in a given basis set. For the mathematical details, see e.g. JA Pople, PMW Gill, BG Johnson Chem Phys Lett 199, 557 (1992).

To answer your questions in this context:
1) The initial density depends on the guess, which is a whole different can of worms, but the same is true for HF calculations. (Procedures based on tabulated atomic densities exist, but the initial guess business is a bit of a dark art, and I don't think this is what you meant by your question.) The basis set and its MO-like coefficients (defining the KS determinant) are used on every iteration to yield the density.

2) Naturally, yes. The MO-coefficients of basis sets change on geometry change for a given system, just like in HF. The procedure and workable algorithms are very similar to HF (because it all is).