The Hulk's basics consist of being able to lift 100 tons, leaping several miles as a form of travel, being able to run at mach 7 speeds, and is incredibly resistant to extreme forms of physical damage.
Is it possible through genetic engineering that a creature like this can come to be? What would be the science to how a creature like this would function?
Possible? sure.
Probable? No.
Each of these characteristics would demand incredible expenditure of energy and resources that will have almost no return to the creature.
In nature creatures usually evolve to better use the resources they have avaible and fill specialized niches. So you need to think of what kind of energy/food source would require of a creature to be at the same time this strong, fast, and resistant.
Take a cheetah for example, they are the fastest land animal on Earth, achieving over 90km/h, but they only can do that trough specialization and in dentriment of other attributes.
Let's go point by point, right?
So let's start with strength, lifting 100 tons is already way in excess of what any animal can lift, even an elephant can only lift about 7 tons and that's an herbivore that has specialized to be big and strong. If you want a predator you can look at tigers, they are fearsome predators, but even then can only carry about half-ton.
Why is the predator so weak compared to the prey? Because tigers evolved claws and fangs, these things make it so the tiger don't need to apply so much force to cause damage to its prey.
Speed.
What airplane? Gossamer Albatross is a human-powered plane with a top speed of 29 km/h (18mph).
So that's pretty easy! We have this part covered.
Regeneration?
Eh, for most animals it's incredibly expensive resource wise to regenerate, it's not without reason that most animals prefer to simply use scar tissue and be done with that. Now, some creatures can regenerate, but its a slow process.
Resistance?
What kind of enviroment this creature lives that would regularly suggest her to nuclear blasts?
And of course, after all that you end up with the Duck.
A creature that can do everything, but can't do any of them remarkably well.
If this enviroment has evolved this creature, then others will have evolved specialized forms will seem much more impressive.
Okay, so, you ask a totally different set of questions in the body of your post vs the title.
Yes, a biological organism can (and does) do those things, but it's nothing like the Incredible Hulk.
As others have pointed out, the energy expenditure of the Hulk is absurd. This link to a comicsvine post about some of Hulk's more ridiculous feats shows how there's no way he can exist in a universe with our physical laws. Even in the milder version you're talking about, the Hulk lifting 100 tons the three meters or so it would take to put it over his head would take a minimum of 7000 kcal, and that's not accounting for his own mass. He'd have to all but live in a McDonalds just to be able to walk around.
What we could have is something that does all those things more efficiently -- something that can lift hundreds of tons, but doesn't move it around much; something that can propel itself quickly, but only when it needs to; something that...
Well, let's cut to the chase. It's a fungus. By spreading underneath the ground, large fungal colonies lift entire forests; they spread by launching spores with extreme speed; they repair themselves rapidly; and while bullets might not bounce off of any fungus (that I know of), they're very much like the Hulk in that radiation only makes them stronger.
Hopefully, someone comes out of this with a new superhero idea -- although come to think of it, it might just be Man-Thing.
