To the first, yes. For example http://en.wikipedia.org/wiki/Lema%C3%AEtre_coordinates is one.

To the second, when we talk about "coordinates" in GR, we're talking about parametrizing a volume of space-time. Given one parametrization, it is possible to create another parametrization that blows up at any particular spot - not because space-time is weird there - but because you picked a bad coordinate system.

The existence of parametrizations that carry through the event horizon demonstrate that the singularity at the event horizon in Schwarzschild's original metric is not a real problem for GR.

However the singularity at the middle of the black hole is a spot where GR breaks down. It is widely believed that a theory that combines GR and QM will make that singularity impossible.

According to general relativity those coordinate systems are equivalent. Well apart from the fact that the "classical" coordinate system isn't able to describe the inside of the black hole. For an example see the Kruskal coordinate system [1].

[1] https://en.wikipedia.org/wiki/Kruskal%E2%80%93Szekeres_coord...