What are the building blocks of energy? What does energy consist of? Is there 1 fundamental, theoretical particle or something similar that causes energy, just like higgs boson?
Energy is a continuous variable defining four momentum space ( p_x,p_y,p_z,E) analogous to the four dimensional space-time continuum (x,y,z,t). These variables are used to define the equations that describe nature as we know it, up to now. They are all continuous in the formulations we have, (though there are adventurous and highly respected theoreticians who might reduce them to a binary form at an elementary level). Still 0 energy or space dimension is accepted .
If there exists a 'building block' of energy, how can objects smaller than that building block exist? It couldn't have energy itself, could it?
There is none.
The only example I can think of are strings in string theory; they are considered to be fundamental, so do they exist in an energy-less microscopic world? Is energy an inherent property of strings? Or does there not exist an answer to this question?
Strings, like all model theories of physics, exist as solutions of equations in the four dimensional space-time and four momentum space too, no matter their extra dimensions. All are continuous anyway, no fundamental thingy is there.
If was just thinking about it, and I'll add a question which is intertwined with question
How does the Higgs boson get mass when it is the particle that causes mass (from what I've understood)?
This is an unfortunate confusion. It is the Higgs field that gives masses to the massive particles, and is a necessary field in the theoretical Standard Model, which describes all the known elementary particle data up to now. The Higgs particle that was measured last summer is another particle in the zoo of particles of the Standard Model, necessary for its completion and due to the Higgs field, it is a proof that the Higgs field is there, since it validates the standard model. The Higgs particle also gets its mass from the Higgs field.
The force is just the gradient of the potential energy. So it's not true that the energy is just force multiplied by $r$. It is the integral of the force w.r.t. $r$, which gives you the $1/r$ dependence.
Edit: here "potential energy" and "interaction energy" are used interchangeably.
Best Answer
There is so much detail one could go into, but I will try to point out the most important aspects:
The concept of force is closely related to energy: force can be seen as something which changes the energy of a system by doing work on the latter. In kinematics, work is defined by a spatial integral over the force acting on an object:
$W=\int\vec{F}\cdot d\vec{x}.$
Force is defined as a change in a particle's momentum. Since this implies a change in velocity, it will also change its kinetic energy. Another example would be thermodynamics, where a force can change the internal energy of a system.
Within quantum field theory (QFT), the energy of a particle depends on its interaction with other particles. Such an interaction is a quantum mechanical generalization of a classical force and albeit the classical and quantum cases share certain features, there are crucial differences. For further explanation, see my answer to question 3.
Thanks to the theory of special relativity we know that mass and energy are equivalent and related by the famous formula
$E=mc^2.$
The terms mass and energy are often used synonymously.
To describe the four fundamental forces, we have two theories: general relativity, which is a theory of gravitation (GR); and the standard model of particle physics (SM), which is the theory of the electromagnetic, strong and weak interactions. Classical force laws (Coulomb, Newton) arise as low energy limits of these non-classical theories.
In the context of GR, gravitational force arises as an effect of the curvature of spacetime caused by the presence of energy/mass. The force itself can be considered fictionary and a result of the fact that objects follow the shortest paths through spacetime (geodesics).
The SM is formulated in the framework of QFT, and as such one describes particles in terms of fields. The energy of a particle depends on the presence of other fields (in this case, one speaks of "coupling", the theory is said to be interacting). The concept of a force is generalized in such a way that one now talks about particle decay. Particles decay into others according to certain laws with a certain probability that can be calculated (e.g. beta decay).
There can be motion without force. By the definition of force as a change of momentum, one can imagine a universe consisting of particles moving at constant velocity with respect to each other.
Within QFT, momentum transfer is described in terms of scattering (for which you can calculate amplitudes) and decay (the process of a particle falling apart into other particles has to respect momentum conservation).