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.