The idea of a free energy landscape is directly related to the idea of a coordinate variable (CV) (also known as reaction coordinate). CVs are simply arbitrary projections $\vec{s}(\vec{x})$ of coordinate space $\vec{x}$ to a lower dimensional space. This is commonly done to simplify otherwise intractable problems, where one cannot handle the high-dimensional nature of $\vec{x}$ in a meaningful way.
In the absence of such variables, the free energy $F$ of a system is directly related to the integral of the <PERSON> weight of the energy $E(\vec{x},\vec{p})$ of each combination of positions $\vec{x}$ and momenta $\vec{p}$:
$$e^{-\beta F(\beta)}=\int_{{\vec{x},\vec{p}}} e^{-\beta E(\vec{x},\vec{p})} d\vec{x}d\vec{p}$$
where $\int_{{\vec{x},\vec{p}}}d\vec{x}d\vec{p}$ denotes discrete or continuous summation over all available states and $\beta=\frac{1}{k_B T}$. This equation is very nice, because you can directly see what the free energy actually is: the free energy of a macrostate has the same function as the energy of a microstate. As such, you can think of it as an effective energy of a particular collection of samples in phase space.
While the regular free energy $F(\beta)$ is the effective energy of all possible phase space samples at a particular temperature, you are free to define a macrostate in any way that is convenient for your purpose. One way in which you can do that is to express the free energy as a function of a low-dimensional (usually between 1 and 3 dimensions) CV, i.e. $F(\beta,\vec{s})$. 976
This allows you to obtain a free energy landscape, i.e. a way to assign each combination of CV values an effective (free) energy $F(\beta,\vec{s})$. Formally, this is expressed as:
$$F(\beta,\vec{s}) = \int_{{\vec{x},\vec{p}}} e^{-\beta E(\vec{x},\vec{p})} \delta(\vec{s}-\vec{s}(\vec{x})) d\vec{x}d\vec{p}$$
with $\delta(\cdot)$ being the Dirac delta. In the case of protein folding, if you have a single CV which goes continuously from an unfolded to a folded protein, such a free energy profile (i.e. 1-dimensional free energy landscape) allows you to determine the relative effective energy of each of these intermediate states.
I am not sure I understand the degeneracy question, but I think it's much easier to think about CVs in a continuous space, rather than discrete space (which is the case for the strictly classical protein folding models). In such a setting, every energy level is degenerate to some degree, and the mutiplicity of this degeneracy is determined by the Dirac delta function, which "picks out" the relevant combinations of $\vec{x}$ that satisfy the constraint $\vec{s}(\vec{x})=\vec{s}$ (with some slight abuse of notation for brevity). In all cases, both the positions and the momenta are important, since they are an inevitable part of the summation. 976