Learn Farsi: من اینجا هستم.

hastam is the independent "to be" verb. We use it when talking about existence or location, like "I am located here" or "I am present." The short ending -am is usually used for describing what something is, like "I am happy" or "I am a student."

Since hastam already means "I am," do I have to include the word man?

No, you can drop it! The verb hastam already has the "I" built into it with the -am ending. You can simply say injā hastam (I am here). Adding man at the beginning just makes it a little more emphatic, like saying "I am here."

Is the word injā related to the word for "this"?

Yes! injā is actually made up of two words: in (this) and jā (place). So when you say you are injā, you are literally saying you are in "this place."