Paracetamol does help me with pain, but taking the standard dose actually makes me feel lower than before. I think it's to do with the way my liver metabolises things (it's also particularly sensitive to anti-histamines).
Ibuprofen for me is a more effective painkiller, but I am aware of its gastric side-effects including increased risk of stomach ulcer and bleed. (Heck, NSAIDs are also implicated in an increased risk of coronary events, hence why no-one in cardiology dares prescribe them anymore.) As Bob correctly stated, NSAIDs are bad for pulmonary hypertension as they are (generally broad) inhibitors of prostaglandin synthesis and activation, whereas prostaglandins and their agonism are essential treatments for pulmonary hypertension (e.g. iloprost-based inhalers and medications) in order to keep the pulmonary circulation going.
I've only taken aspirin once or twice before, but for me it's good for upper tract pain including toothache and sore throat. Again, bad if you're prone to stomach ulcers, as it also can predispose to both ulcer formation and (through its antiplatelet function) increased (sometimes serious) gastric and duodenal bleeding.
I've never taken codeine much before, but as an opioid-based analgesic (like morphine) it's pretty good. It's the constipation that can get people down, though, for the most part. I wouldn't recommend it for some headaches - from what I've seen, tension headaches can sometimes be made worse by opioids.