What are the dimensions, width and length, of a photon?

The fundamental particles we know today (of which the photon is one) are called fundamental exactly because they have no substructure, or indeed, spatial extent, we know of. They are point-like when localized.

Note that these "particles" are quantum objects, not classical particles, so you should not imagine them as points whizzing about in space - they possess delocalized states where they take no definite shape at all (for example, the "electron cloud" around atoms is such a delocalized state).

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The above is a short, non-relativistic view of "particles". When going to the relativistic description that is actually needed for the full description of fundamental particles, things get considerably more murky. For one, we lose the naive position operators, and the notion of "localization" becomes a bit ill-defined because the new "position operator", the Newton-Wigner operators, do not allow to speak of localization in an observer-independent way. The generic particle state that is scattered in QFT calculations is usually a sharp momentum state, and therefore strongly delocalized, so any notion of "point-like" can't really rely on the localization of a particle state.

In this picture, the proper notion of a "point-like" particle is one whose scattering behaviour indicates no substructure or spatial extent. For extended objects consisting of subobjects, their scattering behaviour will typically change when the energies/length scales of the scattering process reach their size, because then their internals get resolved and the individual subobjects start participating in the scattering. So then our notion of size becomes that the scattering behaviour is scale-independent. For more on this notion of size in QFT, see e.g. this answer by Bosoneando.