Am I correct to assume that I can copy Ethereum's source code, modify it to create my own blockchain and issue my new currency, without any problems provided that I follow GPL and LGPL?
The GPL allows you to create incompatible forks and to compete with the original implementation, so no problem there. However, you can't pass your changed version off as the official Ethereum implementation:
When you publish your modified version, it “must carry prominent notices stating that you modified it”. (GPLv3 section 5.a)
If “Ethereum” is a trademark, there may be restrictions on your usage unless you remove all branding, but that is outside of my expertise.