Learn Farsi: انگار او اینجا نیست.

You can use که here and say انگار که او اینجا نیست, but it is very common to drop it in everyday Farsi. When connecting thoughts or reporting a situation, که is often completely optional, especially after words like انگار (it seems / as if).

Does the verb tense change if I am telling a story in the past, like "It seemed he wasn't here"?

Not usually! Just like with reported speech, Farsi prefers to retain the original tense of the observation. The immediate reality being observed is "he is not here" (اینجا نیست). So even in a past-tense story, a Farsi speaker will often keep نیست in the present tense to preserve that original thought.

Is انگار (engaar) a verb that I need to conjugate?

No, in modern Farsi it acts as a fixed word. You don't need to conjugate it for different people or tenses. You just place it at the start of your phrase to mean "it seems," "looks like," or "as if."

Since Farsi often drops pronouns, can I remove the word او (u) here?

Yes, you can simply say انگار اینجا نیست. Because the verb نیست (is not) already shows that the subject is a third-person singular (he, she, or it), the word او is entirely optional and is only used here for a little extra clarity or emphasis.