I can only tell by my own experience and yes you do need to do that. You unlock the bootloader to flash a new recovery in order to flash custom ROM/root and backup current ROM and data. I you run an older version of android like 6.1 you might be able to use kingo root software to root without...
If the new phone has micro SD support you can put a 32gb one or higher to transfer everything, otherwise you need to get a new one with more storage or not transfer everything.
You need to reformat to NTFS or FAT32 and i think GPT can also be selected while formating. You should be able to format through freedos or windows x install cd/usb
note that both have no optical drive if you install games trough cd.
select the ssd one IF it is upgradeable to have both the ssd and a hdd (cuz 128gb is very little by the standards of today)
Try using a different mouse. If it works normally, then it's the mouse's problem. If it work just as the first mouse, then it's most likely a hard/software problem with the computer.
You need one with a beefy Processor and 8-32GB of RAM to make it go fast and easy.
These spec are easier with windows computers. I've never owned a mac pc but windows is perfect for me.
If you choose a windows pc i recommend windows movie maker, sony vegas and wondershare video editor.
If you can access the HDD and internet you should install AntiMalwareBytes and SuperAntiSpyware. If you can't properly boot it i recommend getting a copy of Parted Magic 2013_08_10 and install it onto a USB or CD drive.
I don't own a Nvidiashield but teamviewer is often very laggy and only works properly with non-game programs.
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Even if a wizard himself would make a steam game working it is highly likely to be extremely laggy and not worth it in the end.
Seems like it got hard-bricked. How to know it is hard-bricked:
-doesn't boot
-doesn't make any sound
-the screen doesn't notice the pc
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I recommend going to a electronic store with support to make it un-bricked again at a cost off money.