Getting the list of features used during training of Random Forest Regressor
Is there a function which allows to get the list of names of columns
  used during the training of the Random Forest Regressor model?

RF uses all features from your dataset. Each tree may contain sqrt(num_of_features) or log2(num_of_features) or whatever but these columns are selected at random. So usually RF covers all columns from your dataset.
There may be edge case when you use a small number of estimators in RF and some features may not be considered. I suppose, RandomForestRegressor.feature_importances_ (zero or nan value may be indicators here) or dive into each tree in RandomForestRegressor.estimators_ may help.

If not, then is there a function which for the missing columns would
  assign Nulls?

RF does not accept missing values. Either you need to code missing value as the separate class (and use it for learning too) or XGBoost (for example) is your choice.
I used one set of data to learn a Random Forest Regressor and right now I have another dataset with smaller number of features (the subset of the previous set).
Is there a function which allows to get the list of names of columns used during the training of the Random Forest Regressor model?
If not, then is there a function which for the missing columns would assign Nulls?