I don't know much about the LGPL/GPL when it comes to distributing programs (without a Qt license); If I made a program, could I statically link the libraries to the exe so I have a single file to distribute when:
The program does not cost anything, it's just being distributed (closed source or open source)?
When you have to pay for the program can you still statically link?
No, if your program is closed source and you want to link against the LGPL version of Qt you must use dynamic linking. If you want to statically link then you must buy a license for Qt.
To use LGPL code in your closed source project, the user has to be able to replace the lgpl portion of the code. The easiest and by far the most common way to do this is to put all the LGPL code in a dll and then the user can replace the dll if they choose.
You are also allowed to use whatever technical means your language allows to accomplish the same goal. You can distribute object files and then the user can relink as other commenter pointed out, but I have never seen this done in practice.
It does not matter whether or not your product is free or pay. You can sell GPL/LGPL products.