I am puzzled by many aspects of your response, and I wonder if you meant to write about interpreting fixed effects rather than random effects.
With a model such as yours, I would say that we generally are not interested in interpreting the random effects, other than evaluating (either formally or informally) whether the variances/covariances are different than zero.
The scale of the random effects is determined by the scale of the response, not the scales of the predictors. With a gamma distribution and a log link, (co)variances are on the log scale.
We are, of course, interested in interpreting the fixed effects, and then the scales of both response and predictors do matter. Transformations and/or links (here, log for the gamma) change the fundamental form of the relationship between response and predictor; if you are fitting a linear regression model (including a generalized linear mixed model), then some combinations of scales for response and predictors may better meet the assumptions of linearity. Rescaling, such as centering or standardizing, does not affect the form of the relationship.
As personal rules of thumb: Ideally, I would only transform predictors to better meet linearity assumptions, or for some other "sensible" reason. For example, if one of my predictors is a pre-treatment/baseline measure of the same variable as the response, I transform the predictor to match the scale of the response. (I do not think you have a baseline predictor here.)
As for rescaling (as distinct from transformation), I generally center continuous predictors in regression, and I always center continuous predictors in regression when the predictors are involved in interaction. If I am having estimation trouble, or if I want to interpret predictor effects relative to a one standard deviation change, I standardize continuous predictors. Rescaling does not present any complicated challenges to interpretation.
I hope this helps move you forward.
If you have a professor, then you are at an institution where you might be able to find a statistician to provide more intensive help.
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