It does not matter if the population or the sample is normally distributed. It matters if the residuals are. You should run them and see if they are normally distributed before you reject a method for this. Also normal distributions are not so critical with a large sample size because of the central limit theory. You can transform the data to be normal often as well.1. I do not believe that any of the variables are normally distributed in the population. (In a natural population of animals of all ages there will not be more of average body mass, girth or length than animals of extreme values of those measurements)

That is particularly critical with the IV, the error in the DV will end up in the error term. In practice I would suspect this occurs with most research. I don't think you are going to find a method that somehow addresses this type of error. You can do outlier analysis and then consider whether this indicates measurement error in a case [although you won't know for sure if it is]. You can perform sensitivity analysis and determine what would happen if the error is say ten percent off. But if there is measurement error there is. Other than not doing non-exploratory statistics I don't think there are many options here.2. I have measured both the dependant and independent variables so they all have some error

Then you can't generalize to a larger population. You can essentially do a case study and note the limitation of this design. I personally never found this approach very satisfying, but it is common in analysis. You could suggest follow up research using a random sample.3. I did not select the individuals randomly from the population but rather tried (not very successfully) to have the same number of animals from each body mass class.

This is a violation of the regression assumption of independence. It is in fact common and causes heteroskedacity. One solution is weighted least squares. You should look up heteroskedacity for other solutions.4. The residuals will be dependant on the value of X. (The foetuses weighing 20 kg will have a variation in girth of only a few cm, whereas the natural variation in girth of adults weighing 200kg will be far more).

It does not matter if the IV influence each other or not [commonly they will with real data]. It matters if you have multicolinearity which you can test for with various tests. There are disagnostic packages for this on all commerical software. If you do have multicolinearity there are no easy solutions. You can end up getting stuck with a good overall model, but not being able to speak to what individual variables do.5. The variables definitely affect one another. Mass, Girth, Length and Height are all related. Sex and Season are not.

The real key here is to run various diagnostics and see what problems you actually have. I would think all you can do given data limits is do exploratory analysis. Your problem is at heart your design - how you gathered the data and the relationship of variables and no statistics addresses that. At least that is my opinion (other more expert commentators may have other solutions).

How is your dependent variable measured?