Learn French: Je suis en train de classer les directives. - I am in the middle of filing the directives.

It can, but in an office or administrative context like this, it usually means "to file" or "to organize" documents into an order. If you just wanted to say "to tidy up" or "put away," you would use ranger, and if you meant "to sort," you would use trier.

Since we are learning about the causative faire, how would I change this sentence to say "I am having them filed for him"?

You would say Je suis en train de les lui faire classer. Notice that both pronouns (les for the direct object "them" and lui for the indirect object "for him") are placed directly before faire, rather than before the main action verb classer.

Where does the pronoun go if I just want to say "I am in the middle of filing them" (without the causative faire)?

In a standard construction with a conjugated verb and an infinitive, the object pronoun goes immediately before the infinitive it belongs to. You would say Je suis en train de les classer.

I know être en train de means "to be in the middle of." Do I have to use it whenever I translate an English present continuous like "I am filing"?

No, you don't. The standard French present tense (Je classe) already means both "I file" and "I am filing." You only use être en train de to add specific emphasis, highlighting that you are actively busy doing the action right at this exact moment.