Target disk mode iMac 2012

Thunderbolt 1 and 2 are compatible. You will need the Thunderbolt 3 adapter and Thunderbolt cable.

Thunderbolt 3 (USB-C) to Thunderbolt 2 Adapter - Apple

Apple Thunderbolt Cable (2.0 m) - White - Apple

Why do you need to bother with Target Disk Mode if the data recovery service sent you the recovered files on an external drive? If they sent the files on a bare drive, then just get an external enclosure or drive dock.

If you are trying to access the original failed drive, then good luck. If the professional service couldn't recover everything, then there is no way you will be able to do so. Failing drives produce so many hardware errors the OS and most apps are unable to handle the errors.

Normal utilities (even data recovery software) won't be able to handle any errors thrown up by a failing hard drive since data recovery software is made only to recover accidentally deleted data. The only way you will be able to recover any data from a failing drive is to first perform a bit for bit clone of the drive which requires another drive of equal or larger size. Once you create the clone, you can try using the Finder or a data recovery app on the clone to attempt recovery of your data. Sometimes you may need to use Disk Utility First Aid, Disk Warrior, or TestDisk on the clone to repair file system damage, but at least the physical drive with the data is now not throwing up hardware errors.

The only software I know and recommend to perform a bit for bit clone is the free open source command line utility GNU ddrescue. This utility is able to skip past the bad sections of the drive so you can grab the most data, then later go back to attempt recovery from the bad sections of the drive. You need to use GNU ddrescue's logging system so you can resume the clone if it is interrupted for any reason. While there is a macOS version I find that macOS is unable to handle the physically failing drive. I always perform the clone using a Linux boot drive. If you don't have a Linux system, then you can create a bootable Knoppix Linux USB drive using Etcher (Mac/Windows/Linux). Knoppix includes GNU ddrescue.

After you are finished recovering the data you will need to erase the cloned drive so you can re-use it for other things.

I've successfully recovered data from many very bad hard drives using this method. It can easily take several days to even a week to clone a failing hard drive. The larger the drive and the more severe the failure the longer it will take.

FYI, backups are even more important when using SSDs since it is impossible to recover data from an SSD when you accidentally delete or erase it. Also SSDs can fail at any time without any warning signs.

Whenever I encounter a failing hard drive I almost always make a bit for bit clone of the drive using GNU ddrescue as I mentioned in my previous post. This allows me to recover as much data from the bad drive as possible. Even on some very bad drives I've been able to clone all but several hundred MBs of data (some of it may be empty sections of the drive even).

Once you have a clone of the drive transferred to a fully functional drive it is relatively easy to use standard tools & utilities to access the data from the clone since the clone now resides on a working drive which isn't throwing errors every time you attempt to read from the drive. Sometimes it may be necessary to use Disk Utility or other tools to repair the file system on the clone since it may be corrupted due to incomplete due to the hardware issues of the original drive.

A lot of times I will instead clone the failing drive to a raw image file so I can make other copies. Then I will either work from one of these raw image files or "burn" the image file to yet another drive where I can try to repair the file system and pull the data using the Finder or a data recovery app.

You only get one chance at pulling data from a failing drive so it is best to get the data as quickly as possible. Then put the original drive aside since it is now probably ruined and dead and only work from copies of the cloned drive. I always try to work from a second copy of the cloned data so I can start over if I make a mistake. It can be a very tedious and very time consuming process. Plus it requires lots of storage space to store multiple copies of the cloned drive and the recovered data. I would only consider myself a self taught advanced amateur data recovery person. A professional data recovery service has access to a lot more tools and resources.

Trying to recover individual files from a failing drive is much too risky as you keep increasing the damage on the drive making the problem even worse. I'm not aware of any utility (free or paid) which can attempt recovery of individual files from a failing hard drive. This is the only way I know and the only way I recommend to even attempt data recovery on a failing drive. I've been successful even on drives I never thought I would ever recover even a single file.

I don't work with video files so I'm not sure what tools to recommend for repairing them.