How does a photon interact with other objects? How do we know it is a transverse wave if it's amplitude is energy? I am trying to understand how a photon objectively interacts with something (what does a photon do?).  
I understand it is a force carier of electromagnetic force. It is described on Wikipedia as being a transverse oscillating wave. But the amplitude of an EM wave is also described as a watt. How can it be known to be transverse (as opposed to a longitudinal wave) if it's amplitude is defined as energy? 
When a photon (of sufficient wavelength to interact with an electron) interacts with a massive, charged particle, lets say and electron, what happens to the electron? Does it move with the longitudinal wave? If not, what happens? What happens when the wavelength is too big, does it still effect the electron, but like how a slow current effects a ship, too large to feel it? Or are interactions point like, they either happen or they don't? 
Also, is there an amount of time involved (in the electron's perspective)that can differ in the length of the interaction? Do longer wavelengths alter the em field at a point in space for a longer period of time? 
Thanks so much for any insights, I have done a lot of reading, and I feel like I am missing something fundamental here. 
 A: You are missing the fundamental concept that a "photon" is not light as described with classical electromagnetic fields. All the attributes you assign to the photon are from Maxwell equation solutions for classical electromagnetic waves which are very successful in describing and predicting the behavior of light, a macroscopic phenomenon.
The photon belongs to the quantum mechanical regime, it is an elementary point particle in the very well validated Standard Model of particle physics, and its interactions can be described and predicted using Quantum Electrodynamics.
It can be shown that photons build up the classical electromagnetic field, light, but the process is not a simple addition of energies and momenta.
Here is why you are confused:

The electric and magnetic fields are perpendicular to the direction of motion of light, the spin of the photon is +/- in the direction of motion of the photon.
This image on the right displays a photon, which only has spin +/-1 to its direction of motion, and energy h*nu. The superposition of an enormous number of photons  builds   up the polarization of the classical field.
The photon we measure in the lab does not have an electric and magnetic field varying in space as the classical electromagnetic field does. What it has is a wave function. This wave function is a probability distribution which gives the probability of finding the photon at ( x,y,z,t) and is a solution of QED, with complex functions which contain the E and B fields that the classical electromagnetic wave has macroscopically. 
A confluence of photons is in such a superposition that the classical electric and magnetic fields are built up for the whole bundle as far as the quantum mechanical frame goes. It is not a much used concept because classical electrodynamics works beautifully, the emergence of the classical field from the quantum mechanical superposition  of photons can be seen here, but it needs QED.

When a photon (of sufficient wavelength to interact with an electron) interacts with a massive, charged particle, lets say and electron, what happens to the electron? Does it move with the longitudinal wave? If not, what happens? What happens when the wavelength is too big, does it still effect the electron, but like how a slow current effects a ship, too large to feel it? Or are interactions point like, they either happen or they don't?

A photon and an electron, both elementary particles, interact at a point and the interaction is successfully described by QED using Feynman diagrams.
Here is Compton scattering:

Energy momentum and angular momentum are exchanged.

Also, is there an amount of time involved (in the electron's perspective)that can differ in the length of the interaction? 

The Heisenberg uncertainty applies in all quantum mechanical interactions, so there are time intervals.

Do longer wavelengths alter the em field at a point in space for a longer period of time? 

This is a classical fields question and does not apply to the quantum frame of electron photon.
