Interpreting lasso logistic regression feature coefficients in multiclass problem
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I think you are right that feature weights = 0 across all three classifiers mean that the features are not predictive of the outcome.

In terms of comparing magnitudes, I'm not sure the absolute values are really comparable - each model is trying to predict a different target.

However, you could rank the coefficients in some way and then compare the rank of the coefficient across the three models. You would need to choose a robust way to rank the raw coefficients since they would range from positive to negative - e.g. a strongly negative coefficient such as -2 is probably more predictive than a barely positive coefficient of 0.1.