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Please see the following post, and try to do what I suggested in the comments:
Let me know if you're able to sort it out.
Turns out that it was implemented in one of the Windows 10 updates. See the following blog post, search for "run":
Try holding Ctrl and Shift. See:
Please see my reply here:
https://rammichael.com/7-taskbar-tweaker/comment-page-17#comment-3660
The order of taskbar thumbnails, if that's what you mean, is not remembered by the tweaker. While 7+ Taskbar Tweaker allows to reorder the thumbnails, it cannot restore that order when Firefox restarts, since the thumbnails are not identifiable in any way. That's something that needs to be handled by Firefox itself in order to work properly and be persistent between different launches of the program.
I'm not familiar with RemoteApp. The feature you're asking for requires investigation that I'm unlikely to do, sorry. I also believe that it's less related to the tweaker, since it's not about tweaking the taskbar, but about making it run/appear in a specific situation.
Perhaps asking in a forum such as superuser.com will yield some results.
About the source code of 7+ Taskbar Tweaker, it's not available, but again, I don't think that it can be helpful in this case.
Does it work when you run it manually? If it does, it's clearly a system issue which is unrelated to the tweaker.
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Even though it doesn't have much to do with taskbar tweaking, you should be able to achieve this with the tweaker by using the guide in the following answer:
https://tweaker.userecho.com/communities/1/topics/287-ability-to-run-application-or-even-better-trigger-a-media-button-on-clicking-on-empty-taskbar-space