GPL Licensing – Can Software Be Sold by Original Author?

GPL does not restrict usage in any way. Even making money is OK. Only further distribution is subject to the license terms.

To answer the third question first, the fact that your program uses a library licensed with license X does not necessarily mean that your program must also use license X.
For permissive licenses, like BSD, you are completely free to choose a different license for your own code.
When the library uses a strong copyleft license, like GPL, then you can avoid a lot of legal questions and uncertainty by using the same license for your code.

The reason for this is because copyleft licenses require you to distribute the entire product that makes use of the copyleft licensed source under the copyleft license conditions. So, when you distribute your program, the GPL conditions of having to provide source code apply to both your code, the PyQT library and the additional BSD-licensed library that you use.

Neither the BSD nor the GPL license prevent you from asking money for your program, but they also don't forbid anyone to buy the program from you and redistribute it further without asking money for it.

The reason that PyQT is available both with the GPL and a commercial license is to make it possible to write commercial applications based on PyQT without having to release your source code.