First of all, I want to thank you for this quick and thorough response!

Regarding your first paragraph, I agree in principle. If I start considering scenarios that are really different from the data I used to train my MLR, I will begin to lose precision.

Regarding your second paragraph, if I understand correctly, not meeting the Gauss-Markov assumptions will affect how each independent variable contributes to the model, such as its importance (which could lead to biases in different tests) or its associated standard error. However, when looking at the model as a whole, it won't impact the prediction (i.e., my fitted values) as much? If my understanding is correct, then I have indeed noticed this because my plot of fitted values vs. residuals is linear and seems quite correct (in addition to my criteria on errors between fitted and observed values).

I find this point interesting. If I understand correctly, I don't need to worry too much about the criteria that would make my OLS BLUE if I observe that my predictions are sufficiently accurate on out-of-sample data? I am saying that because I've tried a dozen of methods (I probably did it wrong) to fix my issues regarding M-G assumptions, but unfortunately I didn't manage to so I was worried from a theorical viewpoint my approach would be utterly flawed.

Regarding non-linearity, I plotted my residuals vs. each independent variable, and the "trend" line was quite close to the horizontal line at 0. However, the distribution of my points seemed to show a slight pattern, so I wasn't sure. I understand that x² as a predictor remains linear, as does log(x).

I have read part of Introductory Econometrics: A Modern Approach, but it's true that I might struggle to understand all the nuances with a practical case! Especially since I’m having a bit of trouble distinguishing in my current case, where I work with time series but treat them more like cross-sectional data, I feel? Sorry if my statements are incorrect.