Before I begin answering your question, I just need to clarify one key point.

On the notion of "Uniqueness"

The way you use the word "unique" in your question is not correct. "Uniqueness" has a very precise meaning in a mathematical context and is very different from the way you are using it. When $\alpha$ and $C_p$ are of equal ratio, they produce the same solution curve. However, this does not imply that the solution of the PDE is not unique. Uniqueness refers to whether two completely different functions solve the same PDE with the same data (i.e. data = coefficients and initial/boundary conditions). It is more appropriate to say that the problems with the same ratio $\frac{\alpha}{C_p}$ are equivalent problems and must have the same solution, if one exists.

Please keep this in mind as I answer your questions more directly below:

1. Both the steady state and transient PDE's are well posed (under sufficient assumptions on the coefficients & initial/boundary conditions) for both pure dirichlet and mixed boundary conditions. Thus, a solution exists, is unique, and depends continuously on the data. Therefore, you can uniquely determine the steady state solution using only dirichlet boundary conditions. The transient solution can also be uniquely determined from a combination of the dirichlet boundary conditions and the initial condition.

2. You can determine the coefficients of the analytical solution (1D case) by setting up a system of equations. Suppose the boundary conditions of at $x_1$ and $x_2$ are $g(x_1)=g_1$ and $g(x_2)=g_2$, respectively. Then, the coefficients $C_1$ and $C_2$ are uniquely determined by the system of equations

\begin{align}C_1x_1 + C_2 = g_1\\ C_1x_2+C_2=g_2\end{align}

Before I begin answering your question, I just need to clarify one key point.

On the notion of "Uniqueness"

The way you use the word "unique" in your question is not correct. "Uniqueness" has a very precise meaning in a mathematical context and is very different from the way you are using it. When $\alpha$ and $C_p$ are of equal ratio, they produce the same solution curve. However, this does not imply that the solution of the PDE is not unique. Uniqueness refers to whether two completely different functions solve the same PDE with the same data (i.e. data = coefficients and initial/boundary conditions). It is more appropriate to say that the problems with the same ratio $\frac{\alpha}{C_p}$ are equivalent problems and must have the same solution, if one exists.

Please keep this in mind as I answer your questions more directly below:

1. Both the steady state and transient PDE's are well posed (under sufficient assumptions on the coefficients & initial/boundary conditions) for both pure dirichlet and mixed boundary conditions. Thus, a solution exists, is unique, and depends continuously on the data. Therefore, you can uniquely determine the steady state solution using only dirichlet boundary conditions. The transient solution can also be uniquely determined from a combination of the dirichlet boundary conditions and the initial condition.

2. You can determine the coefficients of the analytical solution (1D case) by setting up a system of equations. Suppose the boundary conditions of the PDE at $x_1$ and $x_2$ are given as $g(x_1)=g_1$ and $g(x_2)=g_2$, respectively. Then, the coefficients $C_1$ and $C_2$ are uniquely determined by the system of equations

\begin{align}C_1x_1 + C_2 = g_1\\ C_1x_2+C_2=g_2\end{align}