Is gravity a fundamental law that is taken for granted? The question might seem silly (and, maybe it is ), but I would like to know if  gravity is really something you can not explain in some other terms and have just to take for granted (i.e. it simply works as it works and we, currently, do not know why), kind of like magnetic force?
Or is there an explanation?
Here are some, exact, points that seem puzzling to me:


*

*If we would build a spaceship resembling Earth in shape and size, would it have gravity, just because of mentioned properties (although, as I understand it, the Earth is round, because of gravity)?

*If we would take the Earth and cut off (kind of like from an apple) 1/10 of it the greater part would still have gravity, while the lesser won't?


*

*What about if we would split Earth in half? 

*Or 40% : 60%?


*Are we currently (or, theoretically, will ever be) capable of creating artificial objects, that have a gravity of their own?
 A: Physics models exist in many frameworks, a number of them emerge from an underlying framework, which is considered fundamental.
What you are asking about comes from the classical gravitational model, which is based on a gravity law and the laws of Newtonian mechanics. 
Laws in physics are like axioms in mathematics, necessary to pickup the correct mathematical model that will describe data, and predict new situations.
All  of your questions are answerable within classical gravitational theory.
answers for


*

*All masses interact gravitationally according to the law, $(G.m_1M_2)/{r^2}$. Shapes and densities and in general mathematical analysis allows the calculation of the force.

*The various cuts will follow the law

*ALL masses have a gravitational field
These answers are within the classical framework
Classical gravitation emerges from an underlying General Relativity mathematical model, but that is another story.
A: Gravity is a force that attracts any objects with mass together. Classically, which is all you need to worry about for now, this force is proportional to the mass of the objects (doubling the mass of one of the objects doubles the gravitational force), and inversely proportional to the square of the distance between them (halving the distance between the two objects quadruples the gravitational force).
Originally, this mathematical description of gravity was applied to the planets in the solar system, to give a good explanation of why they orbited the Sun in ellipses. But planets aren't special, they're just big lumps of stuff in space like everything else, so this proportional-to-mass inversely-proportional-to-squared-distance rule should apply to everything else.
And indeed it does! Every lump of matter, no matter how small, attracts every other lump of matter via gravity. You don't feel very much gravity from small objects because of the proportionality to mass - the Earth is much, much bigger than most other things, so the force from gravity is much, much stronger. And you don't feel very much gravity from distant objects (like the stars) because they're very far away.
You can actually see gravity from small objects happen in a lab. A thin fiber is very easy to twist, so applying a very small force to it can cause it to rotate, until the force from the fiber twisting back equals the force applied to it. By measuring how far it rotates, you can measure how strong the force trying to twist it is.

If done right, it's sensitive enough to detect the gravity from some big lead spheres that you put near the apparatus. Everything has gravity!
This should make the answers to your questions clear: a spaceship that has the same mass and radius as the Earth would have the same gravity (because that's what the gravity depends on), splitting the Earth up would give you two lumps of matter, both of which have gravity, and the forces would be proportional to the mass, and we can make artificial objects with their own gravity (since everything has gravity).
Up until the 20th century, this was where people's understanding of gravity ended. It seemed like it was just a magical pull between distant objects with no inner explanation. Then General Relativity was discovered by Einstein, giving a better explanation of how gravity works.
What general relativity actually says about gravity is well beyond the scope of this answer. The very quick-and-dirty qualitative answer that doesn't give much insight but might convince bystanders that I'm not being smugly coy is that mass and energy are the same type of thing, and that energy causes space to curve, and because space is curved an object traveling in a straight line actually falls towards other objects, and that most of the time you don't need to worry about it because it looks very similar to classical Newtonian gravity except near black holes.
A: This is a pretty good and basic question. When Newton was doing his work within gravity, he actually wrote at the end, that he himself cannot describe gravity and that task was then placed on other scientists; I am paraphrasing of course.
Eventually, Albert Einstein came up with the general theory of relativity in 1916, where he described space and time as a blanket, and anything with mass curving this blanket. Imagine you and three other friends were holding the four corners of a single blanket, each holding a separate corner, to create a tight sheet. If you drop something in the middle, let's say a bowling ball, then the blanket gets a depression in it, where the ball is. Then if you drop a marble at the edge of the blanket, then it will fall toward the larger mass, the bowling ball. This is a very basic explanation of how gravity works.
So, anything with mass has gravity, including you, but the force is so minute you never notice it. And the farther you are away from this mass, the less the force of gravity has on you. So if you were to create a spaceship the same size and mass as earth, then it would have the exact same gravity. If you were to cut off parts of earth, they will both have gravity, of course, the larger one will have more gravity. As mentioned before, everything with mass within the universe has gravity, it's just on a day to day basis the force of gravity of regular items, like a computer,  is extremely small you can never really measure it. To answer your final question, we can theoretically create artificial objects with their own gravity, since everything with mass has gravity, but for it to be noticeable then it must have a large mass. However, we can simulate gravity by using means of centripetal acceleration, or other forces. For example, if we have a spaceship that is spinning, then the outer parts of it will have centripetal acceleration, that can mimic the force of gravity. We have never done this yet though, because of the cost. There is a great video by a youtube channel called, "Real Engineering", that actually covers the topic of artificial gravity.
