So there's a permutation of rotation and translation each being either fixed or free for a column. What do each mean?
To say that a specific degree of freedom of a support is "fixed" is to say that the support cannot translate along, or rotate about, that axis. For example, if it is stated that the translation of a beam is fixed in the X axis at one end, that means that the beam cannot translate, or move, along the X axis (in the direction of +/- X) at that end. And similarly for rotation, if it is stated that the rotation of a beam is fixed about the X axis at one end, that means that the beam cannot rotate about the X axis at that end.
When we design things, we often assume an ideal, or perfect, boundary condition. For example, idealizing a beam as being "simply supported," which means we assume it can rotate about either end and translate along its longitudinal axis at one end, usually enveloped the beam design forces of bending moment and shear force.
In reality nothing we design is ever actually perfectly rigid (fixed) or perfectly free to translate or rotate, we make those assumptions to simplify the design process. In practice, we usually model support conditions that are a combination of fixed translations and free rotations, or vice versa. A very common boundary condition is called "pinned." A pinned support is a support that is not allowed to translate (fixed for translation) but is allowed to rotate freely about any axis (free for rotation).
To answer you specific questions, for an 8x8 post,
If the column is toe nailed to a plywood / beam underneath which is it?
Pinned (free to rotate, not allowed to translate).
If it's in a steel bracket nailed to a column below what is it?
Depends on the bracket and attachment to the bracket, but treating as pinned (free to rotate, not allowed to translate) will usually be conservative.
If it has a 2x6 plate of a wall on top nailed to it what is it?
Pinned (free to rotate, not allowed to translate) .
Does bracing in the middle of the column for either affect it?