Fields
First you need to understand what a field is. There is a very good answer by dmckee on what a field really is which you can (and should read), but I'll try my own version. Mathematically, a field is something that has a value at every point of space and time. A typical example is temperature. The air in your room has a different temperature at every point and this temperature may change with time, so to each point in space and time we associate a number $T$. We might write $T(x, y, z, t)$, indicating that the temperature is a function of $x, y, z$ (space) and $t$ (time).
Temperature is a scalar field because at each point it is a scalar (i.e., a number). But we can have different kinds of fields. For example, the air in your room might be moving around, and so at each point it will have some velocity $\mathbf{v}(x,y,z,t)$. This velocity is a vector field, because at each point it has a magnitude and a direction (if you don't know what a vector is, picture it as a small arrow; the direction tells you which way the air is moving at that particular point, and the length of the arrow tells you how fast it is moving).
Waves
Air can carry waves, which we call sound. Sound is nothing more than a bunch of air molecules oscillating together in such a way that they carry energy from one place to another, in the same way that we see waves in water. With our fancy fields we can describe a wave by saying that at any given point the velocity is oscillating back and forth, and the phase of this oscillation changes as we move from place to place.
Temperature and velocity are fields that, in a sense, don't physically exist by themselves: they describe some property of a fluid, but it is the fluid that has physical reality, not its properties. But there are fields that are not a property of anything else, and the electromagnetic field is the most important among them.
Electromagnetic field
The electromagnetic field is described by two vector fields $\mathbf{E}$ and $\mathbf{B}$, called the electric and magnetic field respectively. For the purposes of light we can forget about $\mathbf{B}$ and just talk about the electric field. Just like the velocity of a fluid, this field can be represented by an arrow at every point in spacetime. Its physical intepretation is that if you place a charge somewhere, there is a force felt by the charge that points in the direction of $\mathbf{E}$ and is proportional to its magnitude. (Also there are magnetic effects but we're ignoring those). This is simply a more sophisticated view of the idea that like charges repel and opposite charges attract; instead of thinking of a force between the charges, we say that one charge creates an electric field near it, which is in turn felt by the other charge.
An electromagnetic wave is simply an oscillation of the electric and magnetic fields. At each point, the field's magnitude is increasing and decreasing with time. Wikpiedia has some nice gifs showing this process in time and space. The wavelength is a physical distance: it's the distance between two maxima or two minima of the field. The amplitude is not a distance, however: it measures how strong the field is, and so it is measured in units of field (Newton per Coulomb or Volt per meter for the electric field in SI units).
You can see in the usual pictures that an EM wave is a transverse wave; that is, the direction of the fields is perpendicular to the direction of propagation of the light. This is in contrast to a sound wave, which is longitudinal: that is, the molecules oscillate back and forth, and the move in the same line that the wave travels.
So, let's answer your questions:
a) The peaks and troughs are the points where the magnitude of the field is maximum in one direction or the other. As such, it doesn't make much sense to distinguish between peaks and troughs, because if you look from the other side they switch places.
b,c,d) A wave doesn't really take up space. There might be fields over a region of space, but the arrows you see in the animations don't have a physical length. They represent the magnitude of the fields, but they don't occupy physical space. Remember that there are two arrows (because of $\mathbf{E}$ and $\mathbf{B}$) at every point in space. As I've said before and has been said in the comments, wavelengths are lengths because they are the distance between two maxima, but amplitudes are not lengths.
The mental picture you describe in your question is, if you forgive me, a mess. You're mixing this description of EM waves with the quantum mechanical point of view, which is almost sure to lead to errors. QM usually deals in terms of particles, so the basic idea is that now light is thought of as a bunch of particles (photons), with a certain probability at each point in space to find a photon. The thing with quantum mechanics is that it's extremely weird and even the very best physicists have trouble forming an intuitive mental image of how it works. So please just forget about photons until you really understand the classical waves I've described in this post.
