QT is distributed under LGPl so by statically linking you must also distribute your program as LGPL. My understanding is that allowing a user to link their own version of QT (ie shared/dynamic libraries are acceptable) fulfils the criteria to avoid licensing your project under LGPL.

>on Windows, you need to copy a dozen .dll files until your executable launches

Presumably you're new to Windows development? as long as you place the libraries in the same folder as the executable, add the QT install directory to your path, or add the install direvtoey to your debigging Environment variable, you're laughing - it's as straightforward as DLLs on Windows come.