How to deal with Nominal categorical with label encoding?
Label encoder is better suited for output classes. You are looking to handle categorical variables. The term is "one hot encoding" which means creating a binary column for each unique value of categorical data. Check get_dummies function in Pandas
It is hard to give any suggestion on model to use without knowing the objective, amount of data available, accuracy requirements. You are better off trying multiple models and checking test set errors to see which one works better.
Variables with high degree of missing values or low fill rate can be removed from the dataset.