Hey there,

Just as the title says, I'm having a hard time with some applications under Plasma5. I am running Ubuntu-16.04 (NOT Kubuntu) with plasma-desktop installed.

By "misbehaving" I mean things like wrong font size (though colours and icons seems to be fine), taking forever to start (e.g. Konqueror) or not starting at all (e.g. Kmix must be called twice to start; the first time it will fail, the second it works).

Any idea where to start digging?

Well, a couple of things I can think of: Kde 4 applications use older libraries which haven't been updated in a while, possibly with fixes for things like scaling, etc. You might check the Qt4 configuation utility about changing the font size on things.

That they use a separate set of libraries means that when they are started there's a good chance that the libraries they use haven't been loaded into memory yet so those have to be loaded first before the application can load. I don't know for sure how much of an impact this has on the startup time though. My initial guess would be that it shouldn't make a notable difference. Another possibility is that the efforts to break up the libraries up in KDE 5 may have resulted in additional savings in startup as there are fewer things which need to be loaded. One more could be from the differences between Qt4 and Qt5.
Also, Konqueror isn't the lightest of applications and likely takes longer to start by itself (which is why it has an option to pre-load instance when you log in).

As for something needing to be started twice, you might compare its config files before, between and after starting it up the two times to see if there is something in there that's being set and/or reset each time it starts.