Cobb-Douglas production function, given $w$ get $r$ regardless of input levels. Why? There is a market economy with technology given by:

$$Y = K^\alpha L^{1-\alpha} \tag{1}$$

Firms behave competitively and input prices are:

$$r = \alpha K^{\alpha-1}L^{1-\alpha} = \alpha(\frac{L}{K})^{1-\alpha} \tag{2} $$

$$w = (1-\alpha) K^\alpha L^{-\alpha} = (1-\alpha) (\frac{L}{K})^{-\alpha} \tag{3}$$

I can solve for $\frac{L}{K}$, and obtain the following expression:

$$r = \alpha (1-\alpha)^{\frac{1-\alpha}{\alpha}}w^{\frac{\alpha-1}{\alpha}} \tag{4}$$

Which is a function that links the prices independently of the inputs combination. 

1) Is this correct?

2) What is the economic interpretation of the last result?
A clarification: The values $w,r$ are not independent of input combinations: you have already solved for $K/L$.

1) Unless you assume that the price of the output is equal to 1, there is a minor mistake, as output prices affect factor prices.

2) If there is a profit maximizing pair $(K,L)$, then for all $\alpha \in \mathbb{R}_+$ $(\alpha K,\alpha L)$ will also be profit maximizing. This is because the production function you are looking at has constant returns to scale (CRS). 

A consequence of CRS is that if the function is differentiable then both marginal products will be homogeneous of degree zero, hence $r$ and $w$ will only determine the ratios of $K$ and $L$, not their level.