Is there an equivalence between information, energy and matter? 
Consider the physical world as made of information, with energy and matter as incidentals.

This statement must be placed in the context of the holographic principle and the works of Bekenstein ...
Indeed, according to Bekenstein

the thermodynamic entropy and Shannon entropy are conceptually equivalent.
the number of arrangements that are counted by Boltzmann entropy reflects the amount of Shannon information that would be needed to implement any particular arrangement ...

...of matter and energy
The only fundamental difference between the thermodynamic entropy of physics and the entropy of Shannon lies in the units of measurement;
the first is expressed in units of energy divided by the temperature, the second in "bits" of information essentially dimensionless.
moreover, the Bekenstein limit asserts that there is a maximum to the amount of information that can potentially be stored in a given region of space that has a finite amount of energy is similar to the holographic principle and that this finally returns to say that the ultimate particle is an information bit (1 or 0).
Can we talk about an equivalence of information, energy and matter ...?
EDIT:
To enlarge the context ,...
Energy and matter
As Einstein showed us, E = MC^2, there is equivalence between mass and energy
Information is energy
Information is a form of energy; indeed to store or transmit information it takes energy
and for his reciproque
energy is information
indeed,
Photons emitted by any source, in the form of radiation such as for example stars, these carry information on the source itself
its location in space and time, its colors and temperatures, its atomic and molecular composition, its speed of rotation and its direction of movement,
..
 A: The first thought to come to mind upon reading this is the Bekenstein-Hawking entropy of a black hole, which relates the entropy of a black hole to the area of its event horizon (which is in turn defined by its mass/energy). If we want to connect this black hole entropy to information, some people have argued that this arises from quantum entanglement. My knowledge of this is rough, but the linked arXiv article may help. 
I link entanglement with information, because when thinking about entangled quantum systems, their entanglement gives us information about the system or lack thereof. 
However, we can also take a classical thermodynamic / statistical mechanic interpretation to the question. In this case we have the quantities of entropy $S$ and internal energy $U$, which can be related (via the first law): $dU = TdS + dW$. This version of entropy is related to the Shannon entropy by a constant factor ($k_B/\ln{2}$?) as $S=k_B\ln{\Omega}$, where $\Omega$ is all possible states that the system can be in. If we want to separate matter from mass, we might talk about individual particles (eg those that make up a gas), in which case their multiplicity (amount of matter) determines the entropy of the full system $S$. $S$ is a measure of how well we know the states of each unit of matter. There's also a competition between entropy $S$ and internal energy $U$, where at low temperatures the system will be in a state where $U$ is minimized but at high temperatures $S$ is minimized. 
Thermodynamics is well defined only in equilibria, but let us consider a system that moves arbitraily from state A to state B. There is a free energy difference between states A and B. The work involved to move the state from A to B can be greater than the free energy difference (eg due to friction). By rewriting the first law, we see that this work lost ("dissipated") must be accounted for by an increase in entropy. So you could equate the work $W$ of driving a process to an increase in uncertainty $S$. But in such dynamic processes, the work of defining entropy is an area of active study, so we need to be careful in our words. 
But all of this depends on context. What is the energy or information of interest? In terms of your two direct questions, we see from the classical thermodynamic description that the entropy $S$ is indeed related to average number of bits needed to describe the system (Shannon entropy). The second on a restriction of information density can be implied from Bekenstein-Hawking entropy / entanglement entropy of a black hole (As the size of the black hole is the limit). 
A: A 2019 paper from Melvin M. Vopson establishes some theory for a mass-energy-information equivalence principle and proposes an experiment to test it.
https://aip.scitation.org/doi/10.1063/1.5123794
From the paper:

The estimated mass of a bit of information at T = 2.73K is m_{bit} = 2.91
× 10^{-40} Kg.

