Looking back over your screenshots.....
It appears you are using Disk Utility in Yosemite or older. The internal disk is showing as Read Only. Connection type shows as external which is puzzling. If you select the Base system does the Mount point show as / (slash)?
The last screenshot with multiple untiltled partitions indicate a problem with the drive. One last thing to try.
Assuming you are booted from Recovery....
Open the Terminal under Utilities in the Menu bar.
1. diskutil list
(identify your drive, for example disk0)
2. diskutil unmountDisk force disk0
(replace disk0 with your identifier from step 1)
3. sudo dd if=/dev/zero of=/dev/disk0 bs=1024 count=1024
(replace disk0 with your identifier from step 1)
4. diskutil partitionDisk disk0 GPT JHFS+ "MyDisk" 0g
(replace disk0 with your identifier from step 1, and MyDisk with whatever name you want...usually Macintosh HD for the internal boot drive)
It is possible that your drive has died. The cost you were quoted seems awfully high.There are videos that show you how to do this yourself. In the US we use OWC sales. See OWC videos: http://eshop.macsales.com/installvideos/
You'll need the model year, serial number and model identifier.
- Under “About this Mac” you'll see model year.
- Next Open System Profiler > Hardware
- Copy the Serial number and model identifier
Example:
iMac (Retina 5K, 27-inch, Late 2014)
Model Identifier: iMac15,1
Serial Number (system): XXXXXXXXXXXX
Since you are in the UK, check out this site for purchasing: https://www.flexxmemory.co.uk/
Their sales should be able to help you selecting the correct drive.
Here is a MX300 525GB SATA 2.5" 7mm (with 9.5mm adapter) Internal solid state drive, SSD by Crucial for £129.95 (Inc. Tax) https://www.flexxmemory.co.uk/apple-mac-memory/imac-ram/upgrades-for-2012-2013-a nd-2014-imacs/late-2012-imac13-1/mx300-5…