Is each succeeding dimension time? I have been researching but I have found nothing on this topic (at least after basic google searching and some deeper searching on this site), I'm not sure if this is a common sense thing or it is wrong. I hear many people argue whether the fourth dimension is time or a spatial dimension, why is it not both? From my knowledge I believe that every dimension is spacial, but every dimension higher is time from the perspective of the previous dimension. If you connect the dots between two squares, you get a cube. And if you connect the dots from two cubes, you get a tesseract. So this would mean if I lived in two dimensions, then would the third dimension be time. And for me right now, the fourth dimension is time. Are the dots being connected just time? 
 A: For physicists spacetime has a precise definition. It is a manifold equipped with a metric. At the risk of over-simplifying, a manifold is a thing that has dimensionality (four dimensions for spacetime) and a metric is a function that defines distances between points in the spacetime.
Whether a dimension is timelike or spacelike is determined by its sign in the metric. Relativity uses a metric in which three of the dimensions are spacelike and one is timelike. Extensions to higher dimensions, for example supergravity, may have varying numbers of spatial dimensions but only have one time dimension.
Your argument about calculating the length between vertices in a square, cube, tesseract etc is a good one because this is exactly what the metric tells us. That is, if you have an infinitesimally small four dimensional cube of size $dx$ by $dy$ by $dz$ by $dt$ the metric tells us the distance between opposite corners. However time is not a spatial dimension. In the metric we multiply $dt$ by the speed of light $c$ to turn it into a distance. For example the metric for flat spacetime is:
$$ ds^2 = -c^2dt^2 + dx^2 + dy^2 + dz^2 $$
In this equation $ds$ is the distance between the corners. Note that we have multiplied $dt$ by $c$ to convert it into a length, and note also that the $dt^2$ term has a minus sign unlike the spatial terms. This is what distinuishes time dimensions from spatial dimensions. That minus sign is also what gives us all the weird effects we see in relativity like time dilation and length contraction.
