Why two ( or more) gases have proportional volumes at constant pressure and temp.? (Avogadro's Law)
So, there are many questions, and I'm not sure if I understand you 100 %:

We have Avogadros Law, which applies for an ideal gas at constant pressure and temperature:
$$\frac{V}{n} = k \quad\Rightarrow\quad \frac{V_1}{n_1}=\frac{V_2}{n_2}$$
With $k$ as a constant.

And we have the ideal gas law: 
$$pV = nRT \quad\Rightarrow\quad \frac{p_1V_1}{n_1T_1} = \frac{p_2V_2}{n_2T_2}$$

And this is probably what your first question is about. There are problems when you have a state $1$ and a state $2$, which differ. 

For example: You have a container where everything ($p_1,V_1,T_1,n_1$) is given. Now you double the temperature - and the task is to calculate the pressure. Since the volume and amount of substance is fixed ($V_1 = V_2$, $n_1 = n_2$), you can just rearrange everything and calculate the pressure $p_2$.

In the ideal gas-model, gases do not seperate. I may be misunderstanding you, but if you put 1 mol $\ce{N2}$ and 2 mol oxygen in a container, you will have different partial pressures, but no demixing.

A relevant question was posted on physics.SE, that may help you if you have to dive deeper into this topic.