Model for Panel data with correlated time-invariant variables after days of reading, I start to get a feeling about the different assumptions, tradeoffs and such for different models. But I still can't find the right answer. I asked a question about the choice of model in another post:
Choice of model/strategy for multiple regression analysis, which gave me some direction.
Below is an artificial overview of how my data could look like if set up as panel data. I will have data for about 100 shipments

I want to estimate the effect of ALL the independent variables and use them as a predictor for the dependent variable
My model is then: The temperature of the shipped products = kg of Dry Ice + transport Time + Ambient temperature
The problem:
1) I expect the effect of the independent variables to be correlated with each other.
2) Kg of Dry Ice and Ambient temperature is time-invariant
The problems imply that I can neither use fixed effect nor random effect. What then? 
 A: From what I see from your data and previous question, all products have the same starting temperature, so the data can be collapsed as follows:
ShipmentID  FinalTemp   Days   DryIce   AmbTemp   
   1           -38        7      150      25
   2           -45        4       80      22
   3           -32        6      100      30

I don't see "kg of Products" in your data, so your model doesn't quite make sense. Excluding this, a model such as 
FinalTemp ~ Days + DryIce + AmbTemp

would seem to make sense.
I do not see why random effects would be indicated. Although there are repeated measures, pre- and post-, we would only want to fit random effects if the baseline values differed.
Since you expect the "effect" of the independent variables to differ depending on the levels of other independent variables, then you can fit interaction terms as necessary. You might also want to consider non-linear associations, for which you can add relevant non-linear terms to the model, make transformations to variables and/or use splines. This can be done with reference to the physical theory of the data generation process, by inspecting model residuals, and model fit statistics.
