Your hard drive is failing. Put a SSHD in it.
500 GB SSD Hybrid 2.5" Hard Drive
Mac mini Mid 2011 Hard Drive Replacement
How can I tell? Just a wild guess after 37 years of Apple experience. Thirty-three of those year were doing Apple repairs professionally. Your old drive is still operational some of the time, so it should be recoverable.
You will need a SATA to USB adapter or enclosure such as this one: Silicone 2.5" Hard Drive Enclosure with USB 2.0 Cable
There's a couple of ways of going about this.
You can put the new drive in the enclosure and hook it up. When you plug it in you should get a message stand it needs to be initialized and Disk Utilities will come up if you stay yes. Format GUID and Mac OS X Extended. It will now be ready for a system installation.
If your old drive is doing OK, it probably has an install on it. Right now it doesn't matter what it is but it will save time if it is Sierra. If the drive is not doing well, it's time to be off with old and on with the new.
Don't use any cloning software as this will negate the repair partition being installed on the new drive.
Now that you have the new formatted drive installed you can install a system either from an external USB source or do an internet install ( this method can take a couple of hours). If doing an internet install just boot it up and wait till it sees there is no system. I like to do these with a wired internet connection via the ethernet port to the router.
Install the system. When it finishes do the setup with your Apple ID and password. At about this point it will ask if you want to transfer your data. Plug in your old drive via USB and say yes and then transfer your data.
This can take some time with a failing drive so just let it run overnight.
If all else fails with your application recovery, email me and I will send you a flash drive with the installations you need.
That failing bring it to the un-conference in September, buy me a glass of cab sav, and I'll fix it with all the software you need ;-) I will charge you a hug after completion.
Finally, when it's all fixed, I want you to get jut a standard 2.5" SATA drive to put in the enclosure for use as a Time Machine backup. This drive does not have to be fast or even new. I'll be putting anti-virus on you machine in September if you don't already have it. What this backup will do is prevent anyone from ever hitting you successfully with Ransom ware. If you ever do get hit, all you will have to do is wipe your drive and restore from the backup. This is actually the only real protection from these guys that there is.