It simplifies developing cross-platform native apps. Imagine electron apps, except it uses less resources, and has a consistent look and feel with the rest of the OS. You also don't have to pay if you don't want, but Qt is licensed under the LGPL so you have to be aware of what you can and can't do under that license.
No, you only have to make your proprietary compiled object files available. what lgpl requires is for the user to be able to update the LGPL parts - so Qt in this case and this does not need your code.
Correct. As long a linking is dynamic, Qt doesn't care how/where the Qt libs are distributed. They typically recommend bundling for Windows and using system packages for Linux.
Every Qt application I've ever used has been ... off.
Granted, it's closer than Swing. That's a pretty low bar. We need to either start qualifying this ("it's a good GUI toolkit for a cross-platform one"), or accepting that this is a scale (10 is "actually native", 0 is "just looks like Motif everywhere"). I'd give Qt 8/10 -- it's pretty good, and the screenshots look great but after a moment using it I'll say "wait, I think I know why everything feels just a little bit off".
As a counter to that, you've only got one set of data points, the ones you noticed. Potentially there are dozens or hundreds of well-made Qt apps that you haven't noticed, and so they by definition wouldn't be a part of your dataset.
