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
code?)

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.

The GPL requires all derivative works to be licensed under the GPL as well. If you fork a GPLed software and distribute your fork, then you are required to offer your software under the GPL. This implies that you have to make the source code available.

From the GPL v3:

5. Conveying Modified Source Versions.

You may convey a work based on the Program, or the modifications to produce it from the Program, in the form of source code under the terms of section 4, provided that you also meet all of these conditions:

• a) The work must carry prominent notices stating that you modified it, and giving a relevant date.
• b) The work must carry prominent notices stating that it is released under this License and any conditions added under section 7. This requirement modifies the requirement in section 4 to “keep intact all notices”.
• c) You must license the entire work, as a whole, under this License to anyone who comes into possession of a copy. […] This License gives no permission to license the work in any other way, but it does not invalidate such permission if you have separately received it.

[…]

6. Conveying Non-Source Forms.

You may convey a covered work in object code form under the terms of sections 4 and 5, provided that you also convey the machine-readable Corresponding Source under the terms of this License, in one of these ways:

[…]

You can choose to violate the license, and distribute binaries of your fork without also distributing the source. In the best case, no one finds out because no one is interested in your software since a better open source alternative exists. But then someone does look at your software – and finds it’s incredibly similar to GIMP. Not just like GIMP is similar to Photoshop, but similar down to the way the menu is structured, or what special effects are available. This person writes a curious email to the GIMP team. Which in turn finds that this is in fact a GIMP fork (e.g. by looking at the disassembled binary or by looking at included assets such as textures) and that you are not distributing the software under the GPL.

What are the consequences?

• If you violate the license of a GPL-licensed software, you loose the rights to use and distribute that software:

  You may not propagate or modify a covered work except as expressly provided under this License. Any attempt otherwise to propagate or modify it is void, and will automatically terminate your rights under this License (including any patent licenses granted under the third paragraph of section 11).

• You are performing a copyright violation. GIMP representatives can shut down any attempt of yours to distribute your illegal software, e.g. by using the U.S.’ DMCA mechanism. For example, you might have created a website to advertise and distribute your software. Using the DMCA, your hosting provider could be forced to suspend your website. If you don’t stop there and are intent on repeatedly infringing the copyright in question, the issue could be escalated to a lawsuit (but no one would be interested in that).

• There’ll be a couple of angry blog posts. Not that problematic in itself, except that anyone searching online for your software will only find posts like “PiratedDraw is a cheap GIMP rip-off” instead of your software.

That’s far too much stress to go through. Don’t be dumb and read the licenses of software you use, even more so of software you want to distribute. In the absence of a license, you are not given any rights to distribute the software – by default, all rights are reserved.