Of all the stuff I've tried on spider mites, the most effective treatment is the ordinary garden hose. It's just too bad I can't use it inside (yet!), but I hose down my outdoor plants every other day. Doesn't kill them all, of course, nothing does short of fire.
I had mold on my apple tree seedlings, and none of the sprays I tried worked at all. I mixed a 50% milk and water spray, and that killed the mold after a week of daily spraying.
Not much experience with other pests, but I doubt there's really a single spray to handle them. There are tons of products making that claim, though. Soap sprays are the most effective against the most pests, though the bugs become resistant to a given spray pretty quick.
In my research the most people who claim lasting success use a variety of treatments, soap, plant extracts like citrus and rosemary, alcohol, etc. I see evidence of this in product reviews too. The 5 star reviews often have a grocery list of other things tried that worked a little but didn't last, then suddenly this one product killed them all. It must be magic.