Learn German: Sie ist eben heimlich schwanger. - She is just secretly pregnant.

eben is a modal particle here, often translated as "just" or "simply". It doesn't change the basic meaning of the sentence, but it adds a specific mood: it tells the listener "that's just the way it is" or "we just have to accept this fact." Without it, the sentence is just a neutral statement of fact.

Can I use halt instead of eben in this sentence?

Yes, you can say Sie ist halt heimlich schwanger. In this context, eben and halt are practically identical. Both express resignation to or acceptance of a fact ("she is just secretly pregnant").

Where should I place a word like eben if I want to use it in my own sentences?

Modal particles like eben almost always go in the middle of the sentence. They come directly after the conjugated verb (here, ist) and any pronouns, but before the main descriptive words (like the adverb heimlich and the adjective schwanger).

Does the word heimlich (secretly) have anything to do with Heim (home)?

Yes, historically it does! Heim means "home". Originally, heimlich described things done privately within the walls of one's own home, away from the public eye. Over time, the meaning evolved from "private" to "secret" or "hidden".