Just to clarify:
1. Foreign stability is affected by your neighbors. If you eliminate an unstable neighbor by war, this category might go up. This may not be specifically what is happening for you though. Also certain civilizations may have more unstable neighbors coded in or other modifiers.
2. Yes. You get a penalty that offsets the stability bonus from acquiring that city. HOWEVER, if your civilization is large enough (AP above suggests this is around 25 in RFCE, in RFC it was 15), shedding excess cities will in fact boost your stability as there are penalties on over expanding.
3. Not surprisingly inquisition hurts stability (Except for the Spanish and their unique power). But I believe so does running certain civics with cities that have extra religions. Those Islamic citizens could also protest your righteous crusade against Arabia, which would result in unhappiness which will result in instability. Which religions to maintain can be a delicate balance, but personally, I try to run less oppressive religious civics (Organized Religion, or Free Religion), and enjoy the extra happiness from extra religions.
4. Economic stability is calculated by crop yield, hammers, and gold income. I believe it accounts for gold spent on any of your sliders as well. It also includes your ratio of imports versus exports (The specifics on this one I'm not sure of. You'll have to look into the RFC forums or wiki on that). As AP pointed out, you want to have your total product (Food, hammers, gold) total slowly grow for the best long term economic stability.