In high energy physics the energy scale is very important. As you said matter is probed at smaller and smaller distances, and that requires more energy. Why is that?
Well in natural units ($c = \hbar = 1$) we have some quantities that mix with each other, i.e there is very little difference between them (mainly just a proportionality constant) In particular:
$$ [Velocity] = number$$
$$ [Energy] = [Mass] = [Momentum] $$ and
$$ [Mass] = [Length]^{-1} $$
From this it follows that $[Energy]$ is actually just inverse $[Length]$ hence the smaller the distances probed, the higher the energy scales.
If these above relations seem strange think of them like this. The highest achievable velocity is the speed of light $c$ and we already set that to one by our choice of natural units. This means that any other velocity will range from $0 \leq v \leq 1$ thus its a scalar.
Also, from $E^2 = (pc)^2 + (mc^2)^2$ it follows that $E^2 = p^2 + m^2$. The last one, which is the core of your question follows from the fact that $\hbar/(mc)$ has units of length and in natural units it becomes $m^{-1}$.
To finish up, your last point about gravity follows from the fact that gravitons interact very weakly at the energy scales we are probing because gravity only becomes relevant at extremely small distances of the order of the Planck Length,
$$\ell _{{\text{P}}}={\sqrt {\frac {\hbar G}{c^{3}}}}\approx 1.616\;199(97)\times 10^{{-35}}{\mbox{ m}}$$
This equates to huge energies that we have no access to currently. All this done above is called dimensional analysis.
Edit: To address the Higgs boson part of the answer:
Don't consider the Higgs boson as a fundamental interaction because it is not. The reason we need high energies to produce the Higgs is for a different reason. As others pointed out, the Higgs is an excitation of the Higgs field. The boson itself is very massive. Remember mass = energy. To produce a massive boson you need to supply at least enough energy to produce its mass. This kind of energy is not available in our everyday lives. Only the LHC has enough power to produce energy scales that high. But that doesn't affect other particles interacting with the Higgs field to gain mass.
Edit: Added small talk on gravity to address the OP's question in the comments:
For a subatomic particle, the gravitational effects are extremely small due to their tiny masses. For gravity to become relevant for individual particles we need to investigate them at the planck length scales. But gravity in general is relevant in the universe, and that is because astronomical objects are very massive and their combined mass produces gravitational fields that have observable effects.