Here's to address the second inequality.

First, I am pretty certain you copied the inequality wrong, and it should read (dropping the $L^\infty$ factor in time)

$$ \| |u|^2 \|_{\dot{B}^{-1/2}_{2,1}} \lesssim \|u\|_{H^{1/2}}^2 $$

Second, I am a bit amused when I glanced at the original paper that I can't find anywhere obvious where they specified a crucial piece of information, namely that they are working on spatial dimension 3. (It is not in the abstract nor on the first two pages of the introduction, as far as I can see.)

So here goes the usual dichotomy argument. Write $u = \sum_{k} P_k u$ where $P_k$ is the Littlewood Paley projector to frequency $\approx 2^{k}$, here $k$ runs over $\mathbb{Z}$.

The Besov norm we can write as
$$ \sum_{k} 2^{-k/2} \| P_k (\sum_j P_j u)(\sum_\ell P_\ell u) \|_{L^2} $$
The product properties of Littlewood Paley projectors (since Fourier transform moves products to convolutions), shows that

- When $j > \ell + 1$, then $P_j u P_\ell u$ has frequency support approximately at $2^{j}$
- When $j = \ell, \ell + 1, \ell - 1$, then $P_j u P_\ell u$ has frequency support over roughly the entire ball of radius $2^{j}$.

So we can expand (roughly speaking) the Littlewood Paley sums to read

$$ \lesssim \sum_k 2^{-k/2} \left( \| \sum_{j > k} P_k (P_j u)^2 \|_{L^2} + \| \sum_{j \leq k} (P_j u) (P_k u) \|_{L^2} \right) $$

**First Term**

We use the Sobolev embedding $\dot{W}^{3/2,1} \hookrightarrow L^2$ (in 3 dimensions) to get

$$ \|P_k (P_j u)^2 \|_{L^2} \lesssim 2^{3k/2} \| P_k (P_j u)^2 \|_{L^1} $$

Next we drop the projection and get

$$ \lesssim 2^{3k/2} \| P_j u\|_{L^2}^2 $$

So the first term is bounded by

$$ \text{First term} \lesssim \sum_{k} 2^{k} \sum_{j > k} \|P_j u\|_{L^2}^2 = \sum_j \|P_j u\|_{L^2}^2 \sum_{k < j} 2^{k} \leq 2 \sum_j 2^{j} \|P_j u\|_{L^2}^2 \lesssim \|u\|_{H^{1/2}}^2 $$

**Second term**

Start with

$$ \sum_{j \leq k} \|P_{j} u P_k u\|_{L^2} \lesssim \sum_{j \leq k} \|P_{j} u\|_{L^\infty} \|P_k u\|_{L^2} $$

Bernstein's inequality (or the frequency restricted Sobolev inequality, which holds also in the case of $L^\infty$) in 3 dimensions implies

$$ \lesssim \sum_{j \leq k} 2^{3j/2} \|P_j u\| \|P_k u\| $$

So we can write (symmetrizing in $j$ and $k$)

$$ \text{Second term} \lesssim \sum_{j} \sum_{k} 2^{- |j-k|} 2^{j/2} \| P_j u\|_{L^2} 2^{k/2} \|P_k u\|_{L^2} $$

So by Cauchy-Schwarz with weights, we have

$$ \lesssim (\sum_{j,k} 2^{-|j-k|} 2^{j} \|P_j u\|_{L^2}^2 )^{1/2} (\sum_{j,k} 2^{-|j-k|} 2^{k} \|P_k u\|_{L^2}^2 )^{1/2} $$

The two sums are identical after swapping $j$ and $k$, and summing first over $j$ in the second one you get

$$ \sum_{j,k} 2^{-|j-k|} 2^{k} \|P_k u\|_{L^2}^2 = 3 \sum_k 2^{k} \|P_k u\|_{L^2}^2 $$

and so

$$ \text{Second term} \lesssim \sum_{k} 2^{k} \|P_k u\|_{L^2}^2 \lesssim \|u\|_{H^{1/2}}^2. $$