Linear vs Non linear regression (Basic Beginner)
Whether you use a linear model or a non-linear model depends completely on whether or not the data is linearly separable, not the number of features. If you have a dataset that is linearly separable, i.e a linear curve can determine the dependent variable, you would use linear regression irrespective of the number of features. Since real-world data is rarely linearly separable and linear regression does not provide accurate results on such data, non-linear regression is used. 

And to answer your second question, yes you are correct. Overfitting occurs when your model has a very high accuracy for training data but a considerably low accuracy for validation/test data. It happens because the parameters are adjusted completely based on your training set, so there is not a generalised curve. Hence on newer data, the model would perform badly.