Is the linearity of Y and X an assumption for linear regression? There are many posts regarding linear regression, so I'm sorry I'm still coming back to this subject. However, I still have some questions about it. I know for sure that the model should be linear within the parameters. That meaning we cannot have something like:
$$y = \beta_0 + \beta_1^2x_1$$
But I've read somewhere that the relationship between Y and X should be linear. Is this correct? If so, what does that mean? Does this mean that the relationship between Y and all $x$'s should be linear? If that is correct, why would $y = \beta_1 \cdot x^2$ be a linear regression model?
 A: You are right. In statistics linear models are such models which are linear with respect to coefficients $\beta_i$. However there is also "common meaning" of linear, denoting the linear relationship (with random error $\epsilon$) between the variables, for example $y=a\cdot x+b$. 
However many non linear relationships (in terms of $x$ and $y$) can be linearized. For example:
$$y=a \cdot x^b$$
A logarithm $\ln(.)$ can be applied to it and one gets:
$$\ln(y)=\ln(ax^b)$$
Using some properties of logarithms one can rearrange this to:
$$\ln(y)=\ln(a)+\ln(x^b) = \ln(a)+b\cdot \ln(x)$$
Rearranging this finally gives:
$$\ln(y)=b\cdot \ln(x)+ \ln(a)$$
which you can present as:
$$Y = A\cdot X+ B$$
where $Y=\ln(y)$, $A=b$, $X=\ln(x)$, $B=\ln(a)$. This means that you just have to recalculate your values of $x$ and $y$ and perform the linear regression to find $A$, and $B$. Then you need to go back to the original model by finding $a$ and $b$:
$$a=e^B$$ $$b=A$$.
So you can see that it is still somewhat a linear model. 
Of course trying to apply a linear model to nonlinear data in a straightforward way will give you a model that is useless. Try to find linear regression of data generated from the function $sin(x)$ for $x \in [0,2\pi]$
A: A short answer:
Yes, the linearity between $X$ and $Y$ is an assumption.
A somewhat longer answer:
Statistical terminology might be confusing. "Linear" in "linear regression" means being linear in the coefficients, but "logistic" in "logistic regression" does not mean being logistic in the coefficients! Logistic regression is just a name, a convention, under which statisticians understand a certain method.
Now, the only explicit assumption behind linear regression is that the errors are normally distributed, with a constant variance. However, if you attempt to fit a straight line to data generated by a non-linear process (like the quadratic one from your question), the errors (deviances of the data from the fitted line) will not be normally distributed.
You can transform your data, by taking $Z = f(X)$, where $f(\cdot)$ is some non-linear function (possibly a vector), and fit a linear function (a line, a plane, a hyperplane...) through such transformed data, $Y = \beta_0 + \beta Z$ ($\beta$ also possibly being a vector). It's a legitimate mathematical trick, and you'd still be linear in both senses of the word.
