Demand for utility function given prices and budget
Regarding the first question, @Oliv suggestion in the comments to formally write down the consumer's problem, budget constraint, Kunh-Tucker multipliers and all, and solve it (carefully -linear objective functions easily mislead when we are used to deal with non-linear ones), is the way to go in order to see how the mathematical set-up will lead you to the answer that stares you in the face:

Our purpose here is to maximize the utility index. Given unitary prices, isn't it obvious that the multivariable function

$$\hat{u}(x_1,x_2,x_3) = x_1+x_2+2x_3$$

will take its maximum value if we purchase only good $x_3$? For any tiny amount $\Delta x_3$ of $x_3$ that you don't purchase, you lose $2\Delta x_3$ in utility terms, while you gain (in utility terms) only $\Delta x_1 + \Delta x_2 = \Delta x_3$ for any combination of $\Delta x_1$ and $\Delta x_2$. Since $2\Delta x_3 &gt; \Delta x_3$ it will be suboptimal to purchase anything else than $x_3$. That was the "plain logic" part, and I hope it answers the "what I don't understand..." aspect of your question. Put the model in math to reconcile this obvious arithmetic with the formal approach.