BSOD 0x00000124 Hardware issue

Either one will be a good choice. No, it will not do anything to your OS unless you install it. It is a Live CD/USB you can run and try it out. nothing you download from the internet will be saved, but you can and use it in that session. It runs out of ram and nothing is written to your hard drive. We use it a lot of times to recover data from hard drives that won't boot and just for testing at times. As long as you don't click install, it will change nothing.

I would back up all my personal files to an external source first. As far as checking for driver issues, I don't know of any way unless you can boot into the hard drive, which I do not suggest you trying on another machine. Do you have an extra hard drive you can put in your machine?

You can't run easy transfer on the old drive and I would not want to anyway. You may be transferring some things you don't want to transfer. You can just copy/paste from one drive to the other or drag and drop anything you want. Also, you say they both have Windows 7 on them. If the Windows 7 on your 'new' drive was not installed on that system, you need to do a clean install.

You can find almost all of your personal files in the Users/yourusername/ folder. There will be the folders for Documents, Downloads, Music, pictures, videos and most everything else. After you install Windows to the new drive, you can connect the old drive and copy everything out. As a suggestion, just do a basic install, download the updates and run it that way for a while just to make sure it runs and runs right before reinstalling all your programs and user files.

Yes, it's pretty simple. The Live CD/USBs run straight from ram and install nothing. The only way you can mess up is to install it on your hard drive. Anything short of that, you're OK.