Hi there James,
Thanks for stopping by the Apple Support Communities! From what I understand, your MacBook Pro displayed an error about the start up disk being almost full, and since then it will not start up fully. You've been able to access OS X Recovery on that device, and have also tried to utilize Target Disk Mode to pull files to another computer. I can definitely understand wanting to get this resolved, and I'll be more than happy to provide some recommendations for you.
To begin, try restarting your Mac using safe mode to see if that allows you to get past the Apple logo. Please note that it does take longer than normal to boot into safe mode. Here are the instructions for that:
Try safe mode if your Mac doesn't finish starting up
This article provides some insight into the screens that display as your MacBook Pro powers on. In certain instances, there are specific troubleshooting steps based on what symbols you are seeing:
About the screens you see when your Mac starts up
In regards to your question about removing specific files using Target Disk Mode, you should be able to drag and drop files into the trash much like you would with an external drive. You could try to open up Disk Utility on your old computer and use First Aid to verify/repair the MacBook Pro. Once you've done that, try again to drag the specific files into the trash and then empty it. Here's a resource detailing Disk Utility:
Using Disk Utility to verify or repair disks
If you do have a recent Time Machine backup of your MacBook Pro, you would be able to restore from that, including your apps. At the very least, any apps purchased through the App Store could also be reinstalled by accessing your purchase history.
Use Time Machine to back up or restore your Mac
All the best!