1) Can someone sell a fork of an open-source software under GPL
without distributing the source? Example: Can I modify a little bit
GIMP, repackage it under another name, and sell it as a commercial
product without giving source code (like Adobe Photoshop) ?
Holy cow NO! That's exactly the sort of behavior that the FSF made the GPL to fight against. This is blatantly obvious, but it sets up the rest of the questions, which have some details we can help you with.
Is this explicitely forbidden by GPL license? (But then, if the source
is not published, how could someone proove it reuses a part of GIMP's
Yes. The owners of the GPL'd code would sue you, and prove it in court in front of a judge and jury. The evidence they would show would include similarities between the programs' outward appearance, file structure, code-size, performance, any other characteristics I'm missing, documentation, as well as your personal background of asking if this was illegal on stack overflow.
2) Same question for online websites.
Depends how you make it online. You can, apparently, modify GPL'd code and let others interact with it online... as long as you don't distribute the software. This operates under the idea that you can take GPL'd code and do whatever you want to it as long as you don't hand it out to anyone else. Some codeshops have their own proprietary code adjustments to the gcc and never release those publicly or privately, but the employees can still use it in-house. (and every update they must recompile with their own code update).
Presumably, you could sell access to that service... but that kind of makes me uncomfortable. And the owners of the GPL'd code might throw some legal action your way if they feel you're violating their license. They may or may not be right. Remember, anyone can sue anyone for anything at any time.