Any development to Mavericks ended with the release of 10.9.5 in October 2013, so I guess you could say that it is stable.
To me, Mavericks is just as stable as Mountain Lion and I am very happy with the performance, but I never use any of the “highlight” features of the release, so I could just have easily stayed with Mountain Lion.
Strangely enough (for me) I jumped onto Mavericks when it was first released (I am usually WAY MORE cautious than that - but I did have backups), and I never experienced a single glitch or problem. But, like anything, your experiences may vary from mine.
I cannot speak to the apps. that you specifically mention, but I do not think that you would have any problems with the upgrade.
I would say, that if you are totally happy with what you have right now, then do not worry about the upgrade unless you absolutely need the functionality that you mentioned with multi displays. Particularly if this system is a production machine that you rely on.
If you do decide to make the change then I would suggest that you have multi backups using different tools, e.g. Time Machine + Carbon Copy Cloner, so that you can switch your system back if you do not like it.
The only issue you will find is that Mavericks is no longer available in the App. Store. So, you need to find a friend that upgraded to Mavericks and either (i) saved a copy of the installer, or (ii) can download a new copy for you, as it would still be in their list of purchases.
I hope it helps…….