If you take any prime ring , then any matrix ring over it is also prime. This is because the two sided ideals of are just matrices over two-sided ideals of . I'm pretty sure you can then show that he product of any two-sided ideals is zero, if and only if one is zero. Or, if you are into more advanced solutions, since matrices over a ring are Morita equivalent (their categories of modules are equivalent), and you can characterize primeness via modules, it follows that a matrix ring over a prime ring is prime. So, that's an example of a noncommuative prime ring.
For your second question, have a look here: noncommutative algebra - Semiprime (but not prime) ring whose center is a domain - MathOverflow