I have four formulas from SR, and need to relate them. Two are the Lorentz Transform, one is a simple formula for velocity, and the fourth is the formula for the addition of velocity in SR.

Yeah, as he said W is NOT v+w. Instead W is x/t. So continuing from
[tex]w = { x-vt \over t - { v \over c^2 } x }[/tex]
divide the top and bottom by t
[tex]w = { \frac{x}{t}-v \over 1 - { v \over c^2 } \frac{x}{t} }[/tex]
[tex]w = { W-v \over 1 - { Wv \over c^2 } }[/tex]
Solve for W and you will get
[tex]W = { w+v \over 1 + { wv \over c^2 } }[/tex]

In this context, v is not x/t. W is. There are three velocities being related. There is w which is the velocity of some "thing" according to measurements made from the primed coodinate system. There is W which is the velocity of that same "thing" according to measurements made from the unprimed coordinate system. And then there is v which is the speed of one of the coordinate systems according to measurements made from the other.