Yes, if something is too dense, it can become a black hole. The Schwarzschild-radius of an object is 2*M*G/c**2, where c is lightspeed, M is the mass of it, and G is the gravitational constant. If the given mass is inside a sphere having Schwarzschild radius as r, it is a black hole. (it's about 8.8 mm for Earth.)
There is no absolute limit, nothing like: 'no area can have density over 10100 kg/m3!', but compressing things too much is impractical for this reason.
A black hole won't cause the collapse of the universe. In fact, there are a lot of black holes out there, and we are still alive.
Yes, black holes (and any masses by some degree) cause gravitational time dilatation.
But your superdense object can have other problems. If it is not dense enough for a black hole, it may explodes from the internal pressure, or, if you can magically keep it together, it has chance to become some form of degenerate matter:
Supporting a small but very heavy object to prevent it from falling into the planetary core is also a difficult task.