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Partial answer: About linear independence, it is true that if $f$ is non 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. Let $$ d_0=\min\{d \in \mathbb{N}: \hat{f}(d)\neq 0 \} \quad n_0=\min\{ n\in\mathbb{N} : n\leq N, c_n \neq 0 \}. $$ Then we have that $$ \sum_{n=1}^N \sum_{{ d \in \mathbb{N} \, : \, dn=d_0 n_0}} c_n \hat{f}(d) = c_{n_0}\hat{f}(d_0) = 0 $$ which is a contradiction. 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 $p=2$.

Partial answer: About linear independence, it is true that if $f$ is non constant then the dilations $f_n(x)=f(nx), n\in \mathbb{N}$ 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. Let $$ d_0=\min\{d \in \mathbb{N}: \hat{f}(d)\neq 0 \} \quad n_0=\min\{ n\in\mathbb{N} : n\leq N, c_n \neq 0 \}. $$ Then we have that $$ \sum_{n=1}^N \sum_{{ d \in \mathbb{N} \, : \, dn=d_0 n_0}} c_n \hat{f}(d) = c_{n_0}\hat{f}(d_0) = 0 $$ which is a contradiction.

EDIT: If you want to consider dilations and reflections, i.e. $f_n(x)=f(nx), n\in \mathbb{Z}$, just write the function as a sum of an even and an odd part $f=g+h$, then $$ \sum_{n=-N}^N c_nf(nx)=0 \iff c_0g(0)+\sum_{n=1}^N(c_n+c_{-n})g(nx) + \sum_{n=1}^N(c_n-c_{-n})h(nx) =0. $$ If $g,h$ are non trivial then you get two equations for "positive dilations" and so the problem is reduced in the previous case.

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 $p=2$. In particular it is not always true that the system of dilations is total in $L^2(0,2\pi)$.

Partial answer: About linear indipendence, it is true that if $f$ is non 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. Let $$ d_0=\min\{d \in \mathbb{N}: \hat{f}(d)\neq 0 \} \quad n_0=\min\{ n\in\mathbb{N} : n\leq N, c_n \neq 0 \}. $$ Then we have that $$ \sum_{n=1}^N \sum_{{ d \in \mathbb{N} \, : \, dn=d_0 n_0}} c_n \hat{f}(d) = c_{n_0}\hat{f}(d_0) = 0 $$ which is a contradiction. 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 $p=2$.

Partial answer: About linear independence, it is true that if $f$ is non 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. Let $$ d_0=\min\{d \in \mathbb{N}: \hat{f}(d)\neq 0 \} \quad n_0=\min\{ n\in\mathbb{N} : n\leq N, c_n \neq 0 \}. $$ Then we have that $$ \sum_{n=1}^N \sum_{{ d \in \mathbb{N} \, : \, dn=d_0 n_0}} c_n \hat{f}(d) = c_{n_0}\hat{f}(d_0) = 0 $$ which is a contradiction. 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 $p=2$.

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 $p=2$.

Partial answer: About linear indipendence, it is true that if $f$ is non 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. Let $$ d_0=\min\{d \in \mathbb{N}: \hat{f}(d)\neq 0 \} \quad n_0=\min\{ n\in\mathbb{N} : n\leq N, c_n \neq 0 \}. $$ Then we have that $$ \sum_{n=1}^N \sum_{{ d \in \mathbb{N} \, : \, dn=d_0 n_0}} c_n \hat{f}(d) = c_{n_0}\hat{f}(d_0) = 0 $$ which is a contradiction. 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 $p=2$.