Are gravitation and the Fine-structure constant related? 
*

*This NewAtlas article Evidence of "modified gravity" in 150 galaxies strengthens dark matter alternative discusses possible anomalies to gravity.


*The Fine-structure constant is likely to be not so "constant"..
I did not find a theory that relates gravity and the Fine-structure constant, is there any? Because if the second assumption is proved to be correct, a variation gravity can be assumed.
And wouldn't this also naturally lead to the missing "wold-formula" which is being searched since so long to unite all fundamental forces?
 A: Brans-Dicke theories allow for the variation of $G$ with the scalar field $\phi$.
Supersymmetric models, for example see this review, can vary any of the fundamental "constants" and you end up with relationships between the "constants." For example see Eq.s 91,92 in section 7 of this classic review by J. Barrow.
Barrow (2003), which reviews theories that vary $G$ and $\alpha$, give a brief history of the idea of varying the charge of the election, or equivalently $\alpha$ (Teller, Gamow, Stanyukovich...). They start in section 3 by defining a model where $\alpha$ varies in time (due to the electric charge varying in time), and discuss cosmological implications (e.g. $\alpha$ is most effected by cosmological expansion). They also state: "We should not confuse this theory with other similar variations. Bekenstein’s theory does not take into account the stress energy tensor of the dielectric field in Einstein’s equations, and their application to cosmology. Dilaton theories predict a global coupling between the scalar and all other matter fields. As a result they predict variations in other constants of nature, and also a different dynamics to all the matter coupled to electromagnetism."

I did not find a theory that relates gravity and the Fine-structure constant, is there any?

Then in section 3.3, they discuss the Freidmann equations of such a model and the effect of also a time varying $G$. It provides a complimentary effect as from $\alpha$: "This type of behaviour can also be found in the presence of time-varying $G$. If a BD dust universe is exactly flat (k= 0) then $G$ will continue to [decrease] forever. Only if there is negative curvature will the evolution of $G$ eventually be turned off and the expansion asymptote to the Milne behaviour with $a = t$ and $G$ → constant."
Lastly, here's a thesis investigating "the effects of matter inhomogeneities on the cosmological evolution of the fine structure constant using the Bekenstein-Sandvik-Barrow-Magueijo (BSBM) theory."
A: Yes, the fine-structure constant is not constant, namely  in the renormalisation process of Quantum Electrodynamics (QED) the fine-structure constant turns out to be scale-dependent -- or in different words, energy-dependent.
When it comes to gravitation, a satisfactory theory of Quantum gravity still does not exist. Assuming that quantised gravitation is a similar Quantum field theory as QED, one will probably find that the coupling of gravitation is also scale-dependent. But as we still don't have such a theory, it is only speculation. It could turn out that a good theory of Quantum gravity requires completely new concepts that make that a scale-dependence is not there. Well, it is nevertheless probable that such a theory will be scale-dependent, but we have no certainty.
However, at least we can say is that a energy-dependent fine structure constant does not change gravity. If it comes indeed to a scale-dependence of Quantum gravity, then it is due the quantisation and renormalisation process (that is completely independent of QED and its coupling constant) of gravity, but not due the fine-structure constant.
