Often my people become unhappy about the unwieldy size of my empire. Is there any way to just destroy the population on an opposing planet, rather than taking control of it?
Not that I know of. You're better off doing research to solve their unhappiness; in the Expansion and Colonization (Down) tree, there are researches to lower Expansion Disapproval and in the Diplomacy and Trade (Left) tree, there are institutions that you can design that, when built in your system, will increase their happiness.
Additionally, certain factions are inherently happier or more discontent. If you run into this problem a lot, consider trying new factions! Take special note of the [faction] affinity, found by hovering over the faction name in the new game screen or the diplomacy panel. These can vary from how the UE and Sheredyn get manufacturing bonuses for tax rates from about 30-100% to how the Hissho get a huge FIDS bonus for successfully invading a planet, how the Sophons get massive research boosts and the Cravers can devour planets. Whichever one you have, it makes a notable difference.