I had this software "Subliminal-messages" by mindofwinner dot com.

It starts automatically with the system.

Usually, I launch Firefox and KeePass soon afterwards. Within a few minutes, sometimes a minute, the application is killed and I have to restart it again to work.

What's the reason behind it and how it can be resolved?

PS: I'm a noob.

Maybe the FREE period is over?
I'm quoting:
I find this message ambiguous: is it totally free to download now and some time you will have to pay to get it (so it's free as long as you have the binary), or does it mean that whenever you download it, it starts as free and then you will need to be charged (so you need a licence at some point)?

I don't think so. I have used this software on Ubuntu in the past and it worked flawlessly then.

Yes, that's my point: the software would work for a limited period so you are experiencing problems only now...
But that's just an assumption, I don't know the software and there is not so much documentation online...

I looked at the website and could find nothing about shareware-like terms of service.

They also have a video that autoplays in a most annoying fashion.

Have you looked at the logs to see if they provide some information about why the application shuts down so quickly after starting? If an application starts and then stops for some reason other than user action, I'ld expect to find an entry in a log file.

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The program starts automatically. If I don't touch it (except start/stop), the program continues to run.

But if I add some more categories to run, it disappears after some minutes.

they seem to have only precompiled ubuntu debs for 14.04. no sources afaics.
which distro are you using, mjdemarco?

Linux Mint 18.3 Sylvia 64-bit MATE edition.

Device - HP 245 G5

Linux Mint 18.3 is based on Ubuntu 16.04, and not 14.04.
my guess is that that's why you're having problems.

Have you tried to find the source code of the program, maybe together with compilation instructions?

I'm a non-programmer.

share-ware on Linux? you do not have to be a programmer to read code, and get a understanding of plan 'English' though you might have to rummage through 1000's of lines of code until you find something that says kill me after x date of x amount of run time, one should still be able to find something referencing to kill the program within the code.

A programmer you say, you are not, but you are a Linux/GNU user, and being so, you have grep to help you and a brain that should be able to come up with key words to use to try and find within the code to see why, and if it is being killed within the program itself.

neither am i but compiling with instructions is well within the scope of my abilities.