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.