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I've never heard of destructive time, so I can't comment on it. It sounds more like you're talking about proper times, though. There are many different ways to define time, and proper times are probably the most useful. The proper time with respect to a particular object is simply the time measured by that object. You would usually have a bunch of different objects moving around all with their own internal clocks. This makes things easier because all of the quantities involved are intrinsic to the physical situation. There's less of a worry about having to construct arbitrary coordinate systems.
The sort of definition used for the time dilation formulae is different. It assumes that there is one universal definition of time picked out by a single inertial observer. Time dilation etc. are really not fundamental parts of the theory, but are derived once certain definitions and constraints are made. So you really shouldn't put any meaning to the fact that those equations make no sense for things moving faster than light. If something is doing that, then an observer could see it moving backwards in time. As I said before, though, that observation isn't unique. Someone moving at a different speed would see the object moving forwards in time (both observers would agree that it is moving faster than light though). It is reasonable to assume that causality is something that can be universally agreed upon, so then this situation would be contradictory. So we say it can't happen. This is much more fundamental than the arguments about mass increase, which are a little misleading.
Also, spacetime is not modified by the speed that you travel. It is also not "moving" in any usual sense. In the absence of gravity, spacetime is fixed as a single four dimensional structure. All of the strange effects that people talk about (length contraction etc) are really just due to trying to force a split between space and time that is completely unnatural to the geometry. It is much better to learn about things from the invariant geometric perspective, but I think this is quite rare in introductory books.
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