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Is is it true (for all cases) that if a transformation (like a translation for example) cannot be written as a matrix equation it cannot be a linear transformation? Thank you.

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It's true if the vector space has finite dimension.

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What does it mean that a linear transformation can be written as a matrix? It's not enough to put numbers in a table according to some rule. What's important is that the linear transformation can then be effected by matrix multiplication.

This, of course, is only possible when the transformation is linear because matrix multiplication is distributive (essentially, another word for "linear").

Conversely, if the transformation is not linear and you're therefore not able to use matrix multiplication to effect it (in the component space), then what's the point of somehow capturing it by some kind of matrix in the first place?

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    Important thing is that to represent linear transformations by matrix one needs to have at most countable (ordered) basis of the vector space (for countably infinite basis one may assign "infinite matrix" in some sense), and if this is the case then there is a linear isomorphism between the linear transformations and matrices. Clearly any matrix multiplication is represents a linear transformation with respect to some basis, it is the converse case that's not immediately obvious – 2017-02-15
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A translation can be written as a matrix. Here is the translation for 3D: $$ T = \begin{pmatrix} 1 & 0 & 0 & t_x \\ 0 & 1 & 0 & t_y \\ 0 & 0 & 1 & t_z \\ 0 & 0 & 0 & 1 \end{pmatrix} $$ Then $$ T (x,y,z,1)^t = (x + t_x, y+ t_y, z + t_z, 1)^t $$ Chaining of transformations is done by matrix multiplication.