Well, it's certainly clear that this has nothing to do with Gloria:
1) it would have been useful to know you were referring to Celtx when you mentioned "a program" in the original post
2) Celtx is not in the repos, it's a stand-alone download, which means it certainly hasn't passed any packaging quality tests, so there's no guarantee it should work as expected. As in windows, third-party software (those not found in repos) is not recommended, and we can't be expected to support it.
3) What makes you think it should update dynamically? What you have described would make sense as Celtx saving the creation date as part of the filename.
4) Nothing you described has anything to do with icon
5) You should post this at the Celtx forums, as anyone there has already installed it. We can't really be expected to compile and install untrusted third-party software just to see if we can reproduce your problem.
6) Don't cross-post (post identical messages in more than one forum topic)
7) You should post first in the support forums as a question, to see if anyone can help you solve it, this forum is specifically for reporting verified bugs, for which you'd need to supply lots more info.
I'm sorry if the above sounds harsh, but there's nothing we can do about it, so I can quite safely say, no, this isn't being addressed by Mint, but it may well be by the original developers, so asking them is much more appropriate.
If you have a question that has been answered and solved, then please edit your original post and put a [SOLVED] at the end of your subject header
Hint - use a google search including the search term site:forums.linuxmint.com