I realise that your question is a few months old. However I'd like to clarify and answer for you or anyone who may come across this in a google search as I did.
Your question is basically 4 questions:
1) Can one use the (largely) LGPL OpenSource QT to edit GPLv2 Source Code?
A. Yes. If you distribute a binary, as per LGPL you must provide a way for the recipient to re-link the binary with their own or different versions of the libraries provided by QT. GPLv2 code CAN be linked to an LGPL library. In this sense they are perfectly compatible..
2) Do you need to distribute any source code?
A. Yes. You must re-distribute your modified GPLv2 Source. It doesn't have to be GitHub, but anyone who gets a binary from you must be made aware of where to get sources.
3) Does using a commercial version of QT release you from your obligation?
A. Yes and No.
No, because GPLv2 does not allow any modifications to be held under any other license but GPLv2, so you are obligated to follow those guidelines irrespective.
Yes, because you no longer need to keep up with QT's LGPL guidelines, you can choose to not distribute any changes you make to a QT Library and/or not provide a relinking mechanism.
4) Can you charge for your effort?
A. Yes. Whether you go OpenSource or Commercial you can charge your client.