Multilabel classification for a learning to rank application
In the multinomial mode, the docs specify that the outputs of coef_ and intercept_ are as you are seeing them: one output for each target class. The underlying model is three logistic regressions, whose outputs are softmax'ed (or with mode ovr, simply normalized).

As to the broader question, since your three output classes are ordered, you might benefit from using that information. Either just perform regression (assumes that the numeric 0,1,2 are meaningful) or use "ordinal regression."