To attempt an answer to your first question:
For $p=2$, the Sobolev space you've defined assigns an infinite norm to the function $f(x_1,x_2)=1$, while the Beppo-Levi SEMI-norm assigns a quantity of zero to this function (as alext87 already noted in his comment). In the way you've defined them, this means that you cannot say that $\|f\|_{W^2} \leq \|f\|_{BL}$. On the other hand, the Sobolev norm contains more terms than the BL semi-norm, and so you find that $\|f\|_{BP} \leq \|f\|_{W^2}$.
If you'd like to show this inequality: $\|f\|_{W^2} \leq \|f\|_{BL}$, then for the above reasons, you will need to make some extra assumptions on $f$. I will now try to explain a sufficient assumption on $f$ which allows an inequality like this to exist.
Let $B=B(0,1)$ be the ball (in $\mathbb{R}^2$) centered at the origin and of radius 1. For convenience, we will assume that $f:B \rightarrow \mathbb{R}$, and that the Sobolev , and BL norm/semi-norm you wrote have been generalized for such an $f$ in the following way:
$\|f\|^2_{W^2(B)} = \int_B \sum_{0 \leq i_1,i_2 \leq 2, i_1 + i_2 \leq 2} (|\frac{\partial^{i_1 + i_2}f}{\partial^{i_1}x_1 \partial^{i_2} x_2}|)^2 dx_1 dx_2$
and similarly for $\|f\|_{BL(B)}$ (only integrate over $B$).
Then, the Poincare inequality (see http://en.wikipedia.org/wiki/Poincare_inequality), states that as long as $f$ has f$, $\partial_{x_1}f$, and $\partial_{x_2}f$ have mean zero, i.e., $\int_B f(x_1, x_2) dx_1 dx_2 = \int_B \partial_{x_1}f(x_1, x_2) dx_1 dx_2 = \int_B \partial_{x_2}f(x_1 x_2) dx_1 dx_2 = 0$, then we have
$\|f\|_{W^2(B)} \leq C \|f\|_{BL(B)}$
For some constant $C$, and for all functions $f$. Here, $C$ is independent of $f$. To summarize, there exists a constant $C$, so that for all $f$ with $f$, $\partial_{x_1}f$, and $\partial_{x_2}f$ of mean zero, we have
$\|f\|_{BL(B)} \leq \|f\|_{W^2(B)} \leq C\|f\|_{BL(B)}$
For unbounded domains, as in how your above question was asked for $\mathbb{R}^2$, the above answer would probably need to be modified. Generally, but I don't know howto understand these things, it helps to look at the case where the domain is the unit ball, since there the relationships are clearer.
For the second question:
I think your definition for the Sobolev space in two dimensions in part 1 is the standard one, except for exact placement of the constants (for instance you removed the 2, which was your choice, no-one told you to put the 2 there!). In most many instances, no-one cares what constants you put where, as long as you have all the derivatives present with a positive constant for each one. If you want to write out the Sobolev norm for 3 derivatives, just make sure to also include all of the partial derivatives of order 3 in your sum above.
Regarding your Beppo-Levi semi-norm, I've never heard it called by this name, but in some circles it seems to be. I've seen it referred to as the homogeneous Sobolev semi-norm in my own very limited circle, but I'm certain this isn't universal (a quick search on google shows both terms being used). The homogeneous Sobolev semi-norm would (to me) usually be denoted by $\|f\|_{\dot{W}^2}$, with the dot standing for homogeneous.
