My question is, what is the domain of such a parameter
Risk aversion coefficient has actually domain $(-\infty, \infty)$. Since $\gamma= -\frac{U''}{U'}$ and this can be arbitrary positive negative or zero depending on exact utility function. To be more specific, $\gamma$ will be negative for risk loving person, 0 for risk neutral person and positive for risk averse person.
The risk tolerance is just reciprocal of the Arrow-Pratt measure of risk aversion $\delta = -\frac{U'}{U''}$. Depending on utility function it could be negative, positive or even zero.
The reason why most papers assume that $\gamma$ is non-negative is that empirically most people have aversion towards risk. Even though, some people might be risk loving or risk neutral, average person certainly is not risk neutral or risk loving so if you use any representative agent model you would assume representative agent is risk averse (unless you are specifically modeling something like gambling behavior or something like that).
Hence by making further assumptions on how reasonable utility function looks like you can further restrict the reasonable range of the coefficient. For example, if you assume that $U'>0$ and $U''<0$ then clearly $\gamma > 0$. There are further assumptions on utility that can restrict $\gamma$ to be above certain values.
dynamic time model, should this parameter remain constant?
This is hard to answer. On one hand it can be argued people's tastes, including risk appetite change over time. On other hand it is questionable how much they change over person's life time.
While making it dynamic might be more realistic its not apriori clear that this is necessary better modeling choice. You should do some thorough literature review on this if it exists. You could also try to make it dynamic and non-dynamic and compare results between the two models. If it qualitatively does not change anything you may just go with the constant version to simplify the model. But do not forget $\gamma_i$ is derived from utility function, so dynamic structure of $\gamma$ has to be consistent with the utility function.
