If your device was set to backup to iCloud then there is a full backup of data & settings. With a new iPhone it’ll offer iCloud backup for recovery. Some info, contacts & calendar for instance as well as some docs can be viewed at iCloud.com
You need more RAM. With only 1 GB of RAM, the OS is leaving little RAM for apps & it is swapping out memory much to often. Looks like this model can access 2 GB, so fill in the second bank with another 1 GB module. When buying memory check that it works with your model, most sites offer products by model. Also, go to System Preferences > Users & Groups > Select your user name > Select Login Items tab & remove any user apps installed, this will eliminate overcrowding of the RAM at startup. Only run apps that you are actively using That should help
My thoughts are similar, but with ADD I would suggest a basic reformat of the drive which should be sufficient, writing to every block will take a considerable amount of time. The reformat will create new directories & everything else will be forgotten as far as the computer is concerned. Install the OS, making sure that you let it update. Install all of your apps whether they are on a commercial CD/DVD or download from the MacApp store they should be clean. See if after the OS & apps are installed if everything is functioning. Then transfer the data, documents, however your files are organized but don't transfer the user libraries, then restart & see if there are any issues. Since you returned the new Mac, I assume that you are using the original Mac that you were having problems with, run the system profiler & check to see if the installed RAM is correct, ie the computer is aware of all of the memory chips, etc. Then try running TechTool Pro or whatever utility that you prefer that checks the internals...
You said most of the screws were missing, if you haven't replaced them Ifixit sells the entire set of screws for the iPhones or try Amazon. If screws are missing, that's a good way to have connection problems & more
How long is it running at 1% & does it automatically shut down at that point ? The lithium batteries have a built in chip that communicates the estimated battery level to the device. Most often devices are charged before the battery is discharged, needing only a top off to 100%. The chip needs to be recalibrated occasionally. Usually letting it run down until the device automatically shuts off & then recharging the battery to 100% is sufficient, but that has to be a single session for the chip to recalibrate. See if that helps Cheers
With Yosemite you should be able to hold [command] + [R] after powering up your Mac & before the startup tune is heard, that's the "bong'. This will startup your Mac on its Reserve drive which is installed when Yosemite was installed. This drive is invisible when not in use. It has a full copy of Yosemite installed on that partition, including Drive Utility. You can boot up on your reserve drive & effect any repairs to the main drive, such as reinstalling the OS