"Can we relatively freely rotate our 4 dimensional coordinate system for the universe's spacetime such that what was space before (time fixed, say at zero) is rotated "into" the time axis to get varying values of the new time coordinate. Or is there some either mathematical or physical obstacle against completely arbitrary rotations that allows us to talk about space alone in some sense?"
It depends what you mean by "rotate". One way to interpret it is as a transformation that fixes the surfaces a constant distance from the origin. In normal Euclidean space, these are spheres $x^2+y^2+z^2=k$, where $k$ is a positive constant, and we get the rotation we are familiar with where any coordinate axis can be rotated into any other coordinate axis. But in spacetime the surfaces are (4-dimensional versions of) hyperboloids $x^2+y^2+z^2-t^2=k$, with $k$ a positive or negative constant, which are split into two families separated by the lightcone. If $k>0$ then the surface is a hyperboloid of one sheet and the distance to the origin is space-like. If $k=0$ then the surface is the cone representing light rays passing through the origin. And if $k<0$ the surface is a hyperboloid of two sheets in the past and future of the origin. Space and time are distinguished, depending on the sign of $k$.
Thus, rotations that preserve the lengths of the coordinate axes ('length' here being the spacetime interval, not just the space coordinate) cannot transform a time axis (with $k<0$) into a spatial axis ($k>0$) because their 'lengths' are different (the squared lengths have different signs). The time axis can point to anywhere on the part of the two-sheet hyperboloid in the future light cone, but it can never cross the cone. Similarly, the spatial axes can point anywhere on the one-sheet hyperboloid 'outside' the light cone, but never cross it.
That's the situation in special relativity, and I think answers the question you meant to ask. But there is a sense in which the very different question in your title can be interpreted so as to (sort-of) give a different answer in general relativity.
Close to a black hole, gravity has the effect of tilting the light cones towards the black hole. (See 2nd picture here for example.) At the event horizon, the entire future light cone is pointing into the black hole, so it is impossible to escape without going faster than light. In the region inside the black hole the radial direction towards the centre is now time-like, and the direction of time flowing outside the black hole is space-like inside. Space and time are still distinct, separated by a light cone, but the cone itself has twisted round so that what was previously a spatial direction (the direction towards the centre of the black hole) is now pointing future-ward.