Earlier quoted context omitted.
You can static-link in all of Qt. Just build Qt yourself. It can strip out all the things you don't need, even symbols from the libraries you do use, so your binary isn't going to be that big.
I do not think it is possible to have a commercial application, you have to pay Qt’s license.
You just have to open your source, that part which depends on Qt. It's not a real problem. But get a commercial license anyway, the cost is small compared to the other costs of developing your program, and you want to be friends with them.
(There's someone on HN who lives on a single-line modification of an open source program. Trust me, source availability of the source code of your client app won't really make a difference.)