Can the dilaton field take on the value at which the strings become absolute free?

What is the nature of the dilaton? The dilaton is just an scalar field that arises in the spectrum of a quantum closed string. You can work out the details of the quantization of a closed string in any string theory textbook. See for example What is String Theory?.

The novelty of the dilaton field $Phi$ is that it controls the interactions between strings and all of their states. An elementary way to argue this, goes as follows:

Consider the Polyakov action for a closed string on a background with massless $G_{mu nu}$, $B_{mu nu}$ and $Phi$ having non-trivial VeVs, given schematically as $$S_{p}= S_{G} + S_{B} + frac{1}{4pi}int_{Sigma}Phi R^{(2)},$$ where $S_{G}$ and $S_{B}$ are the classical actions for the G and B fields and the symbol $R^{(2)}$ in the third summand denotes the Ricci scalar of the string worldsheet $Sigma$.

By hypothesis we are assuming that the dilaton has a non trivial VeV. Isolate the VeV part $Phi_{0}$ on the term $frac{1}{4pi}int_{Sigma}Phi R^{(2)}$ in the Polyakov action as $$frac{1}{4pi}int_{Sigma}Phi_{0} R^{(2)}.$$

Then we expect that the genus $g$ contribution to the S-matrix of this non-linear sigma model to go as $$e^{-chi Phi_{0}}e^{-S_{P}}.$$ On the other hand, and on general grounds, we expect the genus $g$ contribution to the partition function of the non-linear sigma model to contribute as $$g_{s}^{-(2-2g)}e^{-S_{P}},$$ equating the last to equations $$e^{-chi Phi_{0}}e^{-S_{P}}=g_{s}^{-(2-2g)}e^{-S_{P}},$$ then we conclude $$g_{s}=e^{ Phi_{0}},$$ where $chi = frac{1}{4pi}int_{Sigma} R^{(2)}$ is the Euler-characteristic of the string worldsheet.

Is a theory of non-interacting strings just an idealization? Yes, exactly zero coupling its an idealization. The existence of strings forces the appearance of gravity, and gravity is a long range force that couple anything that has energy momentum (in this case the strings). So, at least two strings separated finite distance feel the gravitational potential, and if they scatter with some CM-energy then an effective interaction takes place. However, as in any quantum field theory. You can ideally work computing something by ignoring gravity and supposing that the dilaton has no VeV.