Since electromagnetic energy is carried by photons and moves in forms of waves, does it mean that a single photon when propagating through space doesn't follow the straight path but instead always moves up and down, up and down like a wave.
The term photon belongs to the realm of quantum mechanics. The photon is a fundamental elementary particle in the standard model of particle physics. Electromagnetic energy is defined well in classical electrodynamics and it does move this energy as a wave in time and space.
A single elementary particle propagating through space is mathematically modeled by a wavefunction which is a solution of a quantum mechanical equation. This is a complex number function, it has a sinusoidal form but the only physically measurable effect is the probability of getting a "photon" signal at a specific (x,y,z,t). It is the probability that has a sinusoidal dependence in space time, not the photon, as can be seen in the answer here. The energy of the photon is h*nu, where nu is the frequency of the classical wave which will emerge from a large number of such energy photons.
So it is not possible to talk of a trajectory of a single photon at the microscopic quantum level. It is only macroscopically, when the atomic source is known, and the interaction footprint of the photon is detected on a screen or a camera that a straight line can be drawn which in effect is the optical ray of the classical em wave.
If so another question arises the speed of propagation of light in vacuum is fixed meaning that it will always take the same amount of time for it to travel from point A to point B, but if a photon always moves up and down it will also mean that it travels longer distance than the distance between A and B and so it ill travel faster than light propagates, is it even possible,
No, it is not possible in vacuum. The photon does not propagate as you imagine, and can only be described by its energy=h*nu and its spin direction. It always travels at c.
In the complicated quantized environment of a medium with an index of refraction the way the photon wavefunctions are related to the emergent classical wave, shows that the individual photon paths, which at the microscopic level are always in vacuum and travel with velocity c, can not be an optical ray. An individual photon impinging on a transparent medium will interact by elastic scatterings with the atoms of the lattice and certainly its path cannot be one straight line. In coherence with the zillions of photons in a classical em wave it is better to discuss the classical paths and let quantum mechanics take care of the individual interactions. A true analysis quantum mechanically needs quantum field theory and is unnecessarily complicated.
Quantum mechanics is deterministic, in the sense that the evolution of the probability distribution (i.e wave function) can be obtained rigorously by solving the Schrodinger Equation.
However, quantum mechanics is not "as deterministic as you would wish", in the sense that there is Heisenberg uncertainty principle. The uncertainty principle arises from the fundamental property that quantum observables do not always commute with each other, i.e. generally $\hat A\hat B\neq \hat B\hat A$, and as a result, if a quantum state has a definite value for $A$, it cannot at the same time have a definite value for $B$, and vise versa. This is in sharp contrast with the way we perceive the world works -- for example we say we know the state of a billiard ball only if we know both its position $x$ and its velocity $v$. Well quantum mechanics says this is simply impossible for quantum states (which means for all states in real life).
But things are not that bad. The uncertainty principle says that the uncertainties $\Delta A$ and $\Delta B$ satisfy $\Delta A\Delta B\sim \hbar$. You see that while it is impossible to be absolutely certain about both ($\Delta A$ and $\Delta B$ cannot both be zero), but you are still pretty certain, in the sense that the "average uncertainty" $\hbar$ is really really a small number.
Best Answer
No, the emergence of the classical EM wave from the quantum wavefunction of the photon is not trivial, because a classical EM wave is made up of many photons. In particular, it is not the case that the classical EM wave is the wavefunction of a photon. (Even more particular, it is difficult to even speak of the wavefunction of a photon, since photons usually arise in a quantum field theoretic ("second-quantized") description where the notion of wavefunction does not exist (but is, if you insist on something comparable, replaced by a wavefunctional))
Also, do not speak of "the electron wave". While an electron - like all quantum objects - carries wave-like properties and can be described by a wavefunction (which is not a function as you might imagine it if we incorporate its spin in the sense that it does not take values in the real or complex numbers), it isn't a wave in any classical sense, and also still carries particle-like properties. It is a quantum object, neither fully wave nor fully particle at any time.
Nevertheless, all quantum objects (and hence all things you might decsribe as "matter waves") of course carry energy - the energy that is in their rest mass and the energy that is in their momentum, though the energy of any given quantum state may not be well-defined, but "smeared out" over a range of energies.
To ask how physical objects carry the properties they do is, deep down, not sensible. How does a classical particle carry momentum? By having mass and velocity! But how does it carry mass and velocity? By...um...moving and stuff. How does it move? Um... You get the idea. "Why/How" is a question that can be asked infinitely many times, but only answered finitely many before you hit a point where the only answer is because it seems that way.