Learn French: Je pensais que ton compagnon était célibataire.

This is the French sequence of tenses (concordance des temps) in action. Because the main verb (Je pensais) is in a past tense, the dependent verb (était) must also shift to the past to match it. In direct speech, the thought was: "Ton compagnon est célibataire" (present). But because the thinking happened in the past, est backshifts to the Imparfait (était), even if his relationship status hasn't changed.

Could I use the Passé Composé and say J'ai pensé que... instead of Je pensais?

You usually want the Imparfait here. Je pensais describes a past state of mind or an ongoing belief ("I was under the impression that..."). Using J'ai pensé changes the meaning slightly; it implies a sudden, specific thought that crossed your mind at an exact moment ("Suddenly, it occurred to me that...").

Can I drop the word que like we drop "that" in English?

No, you can never drop que in French when connecting two clauses like this. Even though in English it is perfectly natural to say "I thought your partner was single," in French, you absolutely must include the connecting word: Je pensais que... (I thought that...).

Does compagnon just mean "companion" or is it always romantic?