How to fix disk corruption and reboot loop on an old 2011 MacBook Pro running macOS 10.13.6?

If you indeed can still boot into safe mode, try that, then download and install a cloning tool such as SuperDuper (there is a free version) and try to make a bootable clone to an external drive. Then boot from that external drive and run its Disk Utility to repair your internal drive. You could also erase/format and reinstall the MacOS when booted from that external drive (or clone back to the internal from it).

I have a 2010 MacBook Air, still working and running High Sierra with its original internal SSD (and all original parts, including even its battery, which is at 84% and shows as "healthy"). So I too like to keep older Macs running. But I agree with Allan Jones, it might be time to retire your 2011 laptop. The inability to boot into Recovery would indicate a serious problem with the drive hardware.

Sometimes you need to run First Aid multiple times to fix all the glitches. However, the fact that the drive is already throwing serious error messages can mean that even if you did get it going, it may fail again days later.

Has the drive in this Mac been replaced? If so, with what? Some aftermarket SSDs are not very Mac-friendly.

I restarted in safe mode,...

When you did Safe Mode, did you let the computer sit untouched for 15-20 minutes before testing? I find safe Mode needs a bit of time to finish its housekeeping chores, especially on old slow drives like those that shipped in 2011 Macbook Pros.

In Macbook Pros of that vintage, the hard drive CABLE can cause a litany of oddball problems if worn due to the vibration of transport and normal use. New cables are still available fo your model:

https://eshop.macsales.com/item/Apple/9229751/

If you can do the swap yourself, it might be worth spending $13 plus shipping to try that. It is NOT worth it to pay a tech US$70-90 an hour to make the swap. The cable seller OWC has video instructions here:

Hard Drive Cable Install Video

so you can judge your ability to swap the cable. I've done the swap on an 2012 Macbook Pro and it's not hard.

I enjoying keeping old Macs running and can afford to play with replacing parts. However, at this point I am doubtful I would keep trying to either restore or replace parts on a 2011 model.

Since you can boot into Safe Mode, then you should create a bootable macOS High Sierra USB installer using the instructions in the following Apple article:

Create a bootable installer for macOS - Apple Support

Does this laptop contain its original factory Hard Drive or has that been replaced? To check the health of the drive, run the third party app DriveDx (free trial period) and post the complete text report here using the "Additional Text" icon which looks like a piece of paper on the forum editing toolbar.

How to reinstall macOS - Apple Support

If your Mac can't start up from the built-in recovery system, it might automatically try to start up from macOS Recovery over the Internet. When that happens, you see a spinning globe instead of an Apple logo during startup. To manually start up from macOS Recovery over the Internet, press and hold Option-Command-R or Shift-Option-Command-R at startup.

Computers that can be upgraded to use OS X Internet Recovery - Computers that can be upgraded to use OS X Internet Recovery - Apple Support