Is Linear/Polynomial Regression sensitive to irrelevant features or noise? Are predictive models based off of linear regression and polynomial regression sensitive to irrelevant features or noise?
Will the weights/coefficients for irrelevant columns be automatically adjusted or will it be detrimental to the model? - a nail in the proverbial coffin?
Can I simply stack features without considering their relevance, and rely on the model to automatically discard the useless features, similar to how decision trees work?
 A: The answer is yes, those models are sensitive to noise, but every model is. The strongest signal will be fit first, and gradually lower and lower signal will be fit, until the signal is not longer signal, but noise. This happens with decision trees as well; the strongest variables are at the top of the tree, but at the very end, the split that the tree is making might not be fitting signal but rather noise.
The Noise vs. Signal tradeoff is the Underfitting vs. Overfitting every model faces. You can read more about that by searching for the Bias-Variance tradeoff.
In order to find the good balance of signal vs. noise in Decision Trees, one can set a depth limit. The depth limit determines when you stop, and ideally should be at the point where there is no signal left to fit. In order to find the best depth, you can use Cross-validation.
The same concept exists for linear regression techniques. You can add a regularizer, which usually is the L1 or L2 norm of the vector of parameters your model use. The model will then try to reduce the norm of the vector, effectively using less variables to fit the model. This way, if your model was fitting noise, you could discard unhelpful variables. However, doing this too much could lead to a model not using helpful feature, and finding the best balance between minimizing the error your model makes and minimizing this norm is finding the best tradeoff.
A: As long as you dont overfit, linear/polyomial regression reduces the noise, thats actually the whole idea. Overfitting leads to errors in your model, since the real information contained in your measurements does not support the additional features and so they mostly depend on the noise. Wikipedia on Overfitting
