All your assets are yours, yes, and the pre-ENIGMA source code is completely your property. This is for the same reason that works created in GIMP or InkScape or even MS Paint are yours. Your game is a work you have created in ENIGMA, and probably various other tools. You may license it how you wish.
But that's not the whole story.
When you compile your game in ENIGMA, it is transpiled (a specific synonym for "compiled") to C++. The code it generates still belongs to you (just as the code InkScape or Flash generates still belongs to you). But then ENIGMA links your game's source against its engine. This is where things get hairy.
Because our engine is GPL, your use of it in combination with your source code (ENIGMA-generated or otherwise) subjects you to the terms of the GPL. Thus, you are free to use and modify it locally however you want, but as soon as you release it to the public, you must provide source code. And because the GPL is a viral license, you must also provide all source code that links against it. This means you must provide the source to your game.
It gets worse. The GPL is crafted in such a way as to prevent companies or groups or individuals from providing only obfuscated versions of the source code, which by some definitions, ENIGMA-generated source code qualifies. Specifically, the GPL defines source code to be "the preferred form of the work for making modifications to it." So that means the plain GML/EDL, like the kind you see in your GMK/GMX/EGM.
Other (non-code) assets are easier to license in GPL-compatible ways, and it's easier to work around that.
As Harri pointed out, members of this team have no intention of enforcing the GPL on users who are legitimately distributing games. If one of our contributors is nagging you, you should report that to one of us. Only copyright holders can enforce the terms of the GPL, so if anyone else is nagging you, you may explain or ignore them. That said, it should be sufficient to take us at our word that we will not force you to release your game as a GPL work. However, we are not legally bound by that word unless you get it from each of us in writing for your specific product, which creates potential trust issues. So if you want to play completely safe, the only sure-fire way is to go for the signatures or adhere to the terms of the GPL.
So let's talk about circumventing the GPL, or leveraging it to suit you. Specifically, you mentioned non-code assets.
Since your game as a combined work is covered by GPL, your users are theoretically free to use, modify, and distribute it as they choose. This may sound problematic at first, but it also means that they are free to use proprietary resources with it locally. Theoretically, you can include placeholder art with your game, or include no art at all. You could then distribute those assets separately, under whatever license you please—such as a basic license to use the art with games locally, but not to modify or redistribute it. Thus, while users are allowed to use the art with your game on their machines, they are not allowed to distribute the two as a combined work (they are forbidden from redistributing your assets).
This is one way of preventing legal redistribution of your complete game. However, since there are other copyright holders of the GPL code, those copyright holders are able to demand that you cease to release packages bundled in such a way. This would be exceptionally petty dickery, which you should once again report to the rest of us. This cannot lead to you being forced to GPL-license your art assets, however.
At the end of the day, though, I'd just like to point out that if people don't want to pay for your game, they aren't going to do it. And if they want to reverse-engineer your game, they're going to do it. Yes, source code makes it easier, but it also makes it less of a specialization, which serves to remove the black market from around it as well as the demand for doing so. In my opinion, you'd do just as well to point out that you expect to make $x/copy and that buying a copy is the right thing for users to do. Nothing in the GPL stops you from putting a gentle nag in your game to remind users to support you by purchasing a copy. Honest users will do so, and dishonest users won't, either way. Only a dick would redistribute copies of your game without that nag in them. Just don't overdo it, or it'll become much more reasonable for people to want to do that.
As for other people publishing modifications, it can happen, and as your game grows in popularity, it will happen. If you cease to maintain your game, someone else might make a spinoff that could catch on. The good news is, you can always cherry-pick things they have added to your game—their changes must be GPL as well. The bad news is, if you don't actively do this, or otherwise maintain your code to make it appealing, or do something to maintain presence as the real originator of this game in the eye of the public, then the world will forget about you and move on to other versions. But then, at the point where your game is really that popular, you'll probably have the resources to take your project in all kinds of creative directions, and regardless of your license, ceasing to maintain your game wouldn't allow you to thrive in the current gaming market for very long.
So yeah, you have a lot of options relating to how to distribute and "control" the distribution of your game. But I wouldn't worry about it too much, especially before any serious development or planning begins.
And regarding the bit on modified code, don't worry about that—you're linking against an unmodified ENIGMA. A link to our repository would be more than sufficient.