Space and time as two aspects of the same thing - physically or mathematically? I've read that "space and time are not independent of each other, but are two aspects of a single entity - spacetime". While it's clear to me how they're two aspects of a single entity in the mathematical sense - as coordinates in 4-dimensional spacetime - I'm not sure if they're physically equivalent.
As physical entities, are they also equivalent and aspects of the same entity (spacetime), physically speaking?
 A: Yes. Let's look at the Lorentz transformations in $1+1$ dimensions which tell us how the observations made by two different inertial observers relate to each other:
$$\Delta x' = \frac{\Delta x- v\Delta t}{\sqrt{1-\frac{v^2}{c^2}}}$$
$$\Delta t' = \frac{\Delta t- \frac{v}{c^2}\Delta x}{\sqrt{1-\frac{v^2}{c^2}}}$$
Now, I specifically want to consider a special case that would help me illustrate the point. Consider the case of $\Delta t = 0$. In this case, you can see that 
$$\Delta t' = \frac{- \frac{v}{c^2}\Delta x}{\sqrt{1-\frac{v^2}{c^2}}}$$
So, the time interval between two events in the primed frame is something that was perceived purely as a spatial interval in the unprimed frame. Notice that this crystal-clear case is only for the specific classes of events I chose, namely, for which $\Delta t=0$. However, as the full Lorentz transformation formulae tell us, what is observed as a time interval in one frame gets contributions from what was observed as spatial interval in some other frame (and vice-versa). 
So, the fact that there is no invariant way of deducing as to what part of what we observed as time interval is "actually" time interval, we say that we are observing a unified entity called spacetime and the specific values of the components (i.e., space and time intervals) depend on how we choose to slice the spacetime entity. 
Having said that, there is a distinction between time and space in special relativity. In particular, the measure of this unified spacetime entity is given by $\Delta t^2-\Delta x^2$ which is invariant among all frames. Notice the difference of sign between the spatial interval and the temporal interval (the specific signs are not important--you can flip them, but you cannot change the fact that there will be a relative sign between the spatial and the temporal terms). This tells us that time is indeed on a different footing than space in some sense at least. This is good because otherwise, we would lose any meaningful way of talking about causality. In particular, there is no Lorentz transformation (continuously connected to unity) that can change the relative chronological order of causally connected events. In other words, in no frame did Einstein died before inventing relativity ;) This kind of relative order is not preserved for the spatial coordinates. You can always rotate your frame of reference and make the spatial intervals between events change their signs. 
A: Two observers observe two events that happen. One observer measures them to be separated by a certain distance and a certain time. The other observer, moving relative to the first one, measures them to be separated by a different distance and a different time. There is a specific relationship between these four numbers: the square of the distance minus the square of the time (in appropriate units) is the same for both observers.
This means that space and time are not absolute and not independent of each other. Instead, they are two observer-dependent aspects of a single unified entity called spacetime. The spacetime quantity $(\Delta x)^2-(\Delta t)^2$ combining the spatial and temporal separations is the correct absolute (i.e., observer-independent) physical measure of the separation of the two events.
What directions in spacetime are “space” and what direction is “time” is arbitrary to some degree, in the same way that what direction is “x” and what direction is “y” is totally arbitrary in Cartesian geometry.
A: They are not really equivalent and mathematically they are treated differently. 
In classical statistical mechanics and thermodynamics they are treated completely differently. In quantum mechanics they are again treated completely differently.
Relativity is the difficult one. Time is treated much like a spatial dimension, except that it has a slightly different mathematical expression (with an "imaginary" multiplier). In extreme Relativistic conditions such as near a black hole, time begins to adopt some properties of space and one spatial direction begins to adopt some properties of time, i.e. time starts to become spacelike and space starts to become timelike. Quite what this might feel like is unclear, as anything going near there gets "spaghettified" by tidal forces.
All this has a direct impact on Relativistc quantum theory, in which some aspects of a quantum wave are delocalized not only across space but also across time.
So in some ways they are physically equivalent, in that what can happen with one can also happen with the other, but in other ways they are physically distinct, for example you can change your speed or direction through space but you cannot change your rate of progression through time.
