What I just said doesn't hold true on fair difficulties, because you can't sell resources: AIs don't have money. Thus the feasibility of a longswordmen rush is hindered. On the other hand, you need less units to rush your neighbour, and they will take longer to get on par with you military tech wise. I'd say 3 longswordmen plus fringe units are enough to steamroll your opponent, less if trained (barracks, armory, heroic epic, honor will all help you).
The meritocracy GS will help you get steel a lot faster: bulb it as soon as you get metal casting. Have some swordmen ready for the upgrade. The Great Library may bulb either metal casting (if possible); if you don't think it is, then either civil service for extra growth, or iron working to speed up the process. You will be amazed by its speed. Or just forget about the Great Library altogether and get the National College. Read a thread on the subject if you want to know more. As it is now, NC is stronger (and possibly cheaper) than any other wonder in its era, will get you to metal casting really fast, and won't stop there. It "obsoletes" around Renaissance, but you will have hopefully established a strong lead by then.
For the happiness issues, you have to prioritize techs granting you access to your local lux resources, construction (colisseums and Circus Maximus later on), and Horse Back Riding (circuses). Bribe CS for their lux resources, or fulfill their requests. Monarchy and Representation help with happiness too. And once you get to Theocracy, everything will be fine. If you are going large, FP is a must too. You can dip into unhapiness for a while during your conquests/expansion, just make sure that you have a back up plan, and don't get into the very unhappy range.
8 culture from Stonehenge is great, but CS give as much. They are expensive, but they give you lux and strategic resources, so they make up for it. Plus, they sometimes ask you easy tasks, which lower their cost. And there will be a time (after opera houses, but before Freedom and Radio towers) where Stonehenge won't contribute much towards you total cultural input, while CS culture scale during the game.
I'm not saying your opening is weak. But if you get over your happiness issues (remember, dipping into unhappiness for some turns isn't the end of the world, as long as you know how to get out of there), do an early rush in one of your games, and don't stop till you have rushed ~3 civs, and tell me then what do you think of Stonehenge. Although by then, the patch will have probably arrived, so it won't be the same anymore
