What makes the quarks stay inside the proton? Inside a single proton for example, what is the force(s) that keeps the quarks together? Why don't they leave the proton? If they do, how does that even happen?
And maybe an additional sub question: Is it always three quarks inside the protons/neutrons? Why?
 A: Why do proton's quarks stay together? because they have a force between them called the "Strong Force" studied under what's called Quantum Chromodynamics, about which you can read in Wikipedia. The force is a field quantized in the form of bosons called "Gluons", which work like a glue that works the interaction between individuals quarks.
And about your second question: is it always protons/neutrons? Actually not necessarily. The whole story depends on the life-time of the compounds formed by quarks. A simple example can be given from atoms. So the combinations of protons, neutrons and electrons makes atoms. But does that mean that any combination is possible? Of course not. Some combinations are stable, and some are not. Uranium, for example, exists, but isn't stable, and decays to Thorium. Iron exists, but takes a very very long time to decay.
In other words: You may make many many many different combinations of quarks to form either mesons (particles with two quarks) or baryons (particles with three quarks). Some of them are stable that they have a long life-time (like protons), and some others just decay faster (like neutrons, which have around 880 seconds half-life). And some compounds from Quarks decay instantaneously, like Kaons, that live for around $10^{-8}$ seconds.
