About linear indipendence, it is true that if $f$ is not constant then the dilations are linearly independent. In fact suppose for a finite sum we have \begin{equation} \sum_{n=1}^{N} c_n f_n(x) = 0, \,\, a.e.-x, \end{equation} then the Fourier coefficients must vanish so we arrive at the equation \begin{equation} \sum_{n=1}^N \sum_{{ d \in \mathbb{N} \, : \, dn=m}} c_n \hat{f}(d) = 0, \forall m =1,2,3... \end{equation} and analogously for the negative Fourier coefficients. Since this is a Dirichlet convolution by multiplying with the Dirichlet inverse of the sequence $c_1,c_2,\dots , c_N$ we find that the Fourier coefficients of $f$ must vanish.
The problem about total sets or a Schauder basis I expect to be much more complicated. In the article "A Hilbert space of Dirichlet series and systems of dilated functions in $L^2(0,1)$" of Hedenmalm, Lindqvist and Seip the authors give a characterization of the functions for which $f(nx)$ is a Riesz system (an isomorphic image of an orthonormal basis) for $p=2$.