You misunderstand the result Penrose has shown. There are plenty of aperiodic tessellations of the plane that can be computationally generated. The issue is whether a computer can, in all cases, determine if a set of shapes can tile aperiodically.

I don't understand the second bit. But, no. Computability has very specific definition. Getting a solution to an uncomputable problem isn't generally hard. It is trivial to create a program that will solve the halting problem for an infinite class of cases. The issue is that such solutions can be shown not to be general over all cases.