1. Yes, but when matrix contains only rotations and translations (no scaling).
2. To fully describe transformations in 3D (rotations, translation and scaling) you need 4x4 matrix. Translation must be in 4th row. You will see it if you rewrite the translation eqation
Code:
x' = x + tx
y' = y + ty
z' = z + tz
into the matrix form.

The remaining fourth column contains zeros except the _44 element which is one. Since the matrix is square, it can be inverted.
In order to multiply vector by a 4x4 matrix, the vector must have four components. Usually the fourth one called w is set to one.

The calculations on the picture implicitly assume that w equals one and missing column contains zeros.

As for the projection matrix, the only thing I can say that it contains the view frustum and it transforms coordinates from world space to screen space. It is all I know.