Some questions about feature hashing in the context of document classification
Throughout the hashing process those words may map to the same indices and for the example you gave, i'll say it's extremely likely you'll have collision.  How the hashing table chooses to handle that is up to the implementation.

Feature hashing may purpose is when you have a large amount of data and that data tends to be sparse.  Words tend to be a good example because word matrix of counts tend to be rather sparse.  

The hashing trick is useful for large datasets because the more traditional ways of handling text data is essentially by making two passes on the dataset.  You basically have to go over the dataset once to create a dictionary and then make your transformation.  For large data this can be rather expensive process.  Feature hashing allows for a one time pass of the data.  This also enables the ability to do online learning, which brings me to this point.  I wouldn't say that feature hashing helps with new words rather that it basically handles it for you.   In traditional approaches you have to ignore the new word or create a new dictionary.  Feature hashing will simply just map it to an existing index or a new one if there's space.

The last point is that feature hashing rarely if ever improves performance of a model.  It also hurts the ability to be introspective.  You are going to lose information, and in fact, you can view feature hashing as a way to introduce noise.  However, research indicates that in large sparse data, the noise does not impact performance in major ways.

Hashing Function

Large Scale Hashing

Short idea behind hashing