Here's a few things.
1), check your system settings about phone and see what version you are running. If it is Android 2.3.6 and the info somewhere says .6.13.219. in it you are running the official update. You could still be rooted, check for a SuperUser app in your app drawer. Not an absolute, but usually goes hand in hand with the phone being rooted.
2), game position is a very very lame reason to continue using a device that is as screwy as yours from your description. Not trying to be harsh or mean....but seriously....your phone is acting that poorly and you are more worried about a game position? Factory Reset the device. This is a no brainer.
3), Android will see free available memory and load apps in to them. Not launch them, just hold them in memory. This requires no extra CPU or battery usage (holding a pattern of 1's and 0's takes the same amount of battery and CPU as holding a block of 0's). This is the way Android was designed and will function much better if you do NOT force stop these apps. Most people see them and assume they are running....not true....just loaded in to available memory for faster access by you the user. Some apps (especially Verizon Bloatware) have very high priorities set on them and tend to load up in available memory. You can correct this with root and freezing those but by force stopping them (and you've noticed they are right back up again) you are killing your battery and wasting resources faster as the OS is using battery/CPU to reload those apps in to memory.
4) The phone should never download apps on it's own. Most likely an app you've downloaded has started this process and this is not a good thing. Factory Reset.....start over. Sign in, setup the phone and download apps one at a time. See if you can find the offending app. It also sounds as if your phone is rooted but so you may want to download SuperUser from the Market and see. If the device is rooted but has no Superuser app...any and every application can take root control without asking you for permission....that's what Superuser does for you. If you run Superuser and doesn't allow you to grant root access rights...then you aren't rooted and you can uninstall it. Oh...Factory Reset does not unroot your device.
5) Factory Reset the phone. Save yourself all the hassles you are going through, set the phone up again and start your game over.
I'm not trying to be rude...but something is messing your phone up WAY more than it should. It should NOT get overly hot, especially if not in use. It should NOT charge weird/strangely and it should hold a charge for at least 19 or so hours. It should NOT download apps. It should NOT overly lag and freeze. Eliminate the guess work and Factory Reset. If it continues on just your account with no apps installed by you.......take it to Verizon and swap it out.