Well, a power-off definitely causes all processes to quit. 
When you restart, iOS won’t try to restart everything, even though the list of recent apps (that you swipe-up from the force-quit) will remain populated.
It will restart some apps, but I’m not sure exactly what the rule is. I suspect it will be those that are configured to allow “background app refresh” (Settings → General → Background App Refresh). And even then, I expect it will only launch those parts of the app needed to perform background app refreshing, not the rest of the app.
There are two things that I’ve found can make a phone hot.
One is using it in bright sunlight. Apple’s displays can get very bright, to allow visibility in direct sunlight, but when they’re this bright, they get hot and drain your battery. If used for too long in this state, they can actually trigger an overheating situation, which will cause the display to switch to minimal brightness and slow the CPU a lot.
The other is an app that’s drawing a lot of CPU/GPU power. Games are notorious for this. I’ve found that some ad networks that embed in other apps and web pages can also cause these kinds of problems.
If you’ve got an abusive/spyware app (I’m looking at you, Facebook and Google) that tries to monitor everything you do and phone home with it, that could easily do it as well.
I suspect this is your problem, because restarting the phone temporarily fixes it.
If you can’t identify the problem app, I would suggest that you go to the Background App Refresh page and disable the feature for everything you don’t actually want to operate in the background.
Some apps I leave enabled for backgorund refreshing include:
- My bank apps. I always want them to notify me if something suspicious happens, and I want their content to remain up to date.
- Books. So I don’t have to launch the app in order to get new books that I load from my Mac.
- Notes. Again, so the notes will remain in sync at all times. Especially important because I sometimes need to access them when I don’t have network connectivity.
- Siri. Not sure I need it, but it seems appropriate.
Just about everything else has background app refresh disabled. There is no reason for my games, YouTube, shopping apps, etc. to sync their content when I’m not actually using them. They sync just fine when I launch them, thank you very much.
The other thing I would suggest is limit what can access your location in the background, since GPS and the database lookup associated with it can consume battery and network bandwidth. Go to Settings → Privacy & Security → Location Services and configure each app.
In general, I permit only my credit card apps the ability to access my location in the background (the “Always” setting). I do this because many cards these days will use the location of your phone (once it is logged in to your account) as a part of the validation process. If you travel, charges away from home will be less likely to get flagged as suspicious if the app’s reported location is the same place you’re using the card.
Most other apps have location services set to “When Using”, meaning they will only have access when the app is running in the foreground. And only those apps where I think the location may be useful (e.g. Maps, Weather, Camera, etc.). Stuff where my location should not be necessary (e.g. games) are set to “Never”.