Why is energy always related to time in physics.
I don't think it is helpful to describe energy as "always related to time" in physics.
That being said, there certainly are a number of interesting relationships involving energy and time, such as
$$
\frac{\partial S}{\partial t_f} = -E(t_f)\;,\tag{1}
$$
and
$$
\frac{\partial S}{\partial t_i}=E(t_i)\;,\tag{2}
$$
where $S$ is the classical action.
I won't prove the equalities in Eq. (1) and Eq. (2), but they can be motivated by considering the free classical action:
$$
S_0 = \int_{t_i}^{t_f}dt\frac{1}{2}m {\dot x(t)}^2 = \frac{m}{2}\frac{(x_f-x_i)^2}{t_f-t_i}\;,
$$
for which Eq. (1) and (2) clearly hold.
When it comes to translations and rotations, I think I have a good grasp as why it is like that. For example, I understand why a translational invariance implies a conservation of momentum etc...
Translational invariance in space implies conservation of momentum. Translational invariance in time implies conservation of energy.
But when it comes to energy and time I don't have a clue of why it is like that.
If your Lagrangian does not depend explicitly on time then we have
$$
\frac{dH}{dt} = -\frac{\partial L}{\partial t} = 0\;,
$$
so energy is conserved. (I will not prove the first equality above since it is well known.)
This is one formulation of time invariance (Lagrangian doesn't explicitly depend on time) implying that energy is conserved.
But probably is it better to think in terms of the action, where the invariance in time more clearly leads to the conservation of energy. (See below.)
Another good way to think about how time translation invariance leads to energy conservation is in terms of the action:
$$
S(t_f, t_i) \equiv \int_{t_i}^{t_f}L(x(t),\dot x(t)) dt\;.
$$
Translational invariance of the action with time means that
$$
S(t_f+\delta t, t_i+\delta t) = S(t_f, t_i)\;,\tag{3}
$$
which in turn implies that
$$
\frac{\partial S}{\partial t_f}+\frac{\partial S}{\partial t_i}=0
$$
$$
=-E(t_f)+E(t_i)\;.\tag{4}
$$
Or, rearranging the terms in Eq. (4), we have:
$$
E(t_f) = E(t_i)\;,\tag{5}
$$
In other words, time-translation invariance (Eq. (3)) implies that energy is conserved (Eq. (5)).