Proof of Pitt's theorem
We assume that $\{x_n\}$ doesn't converge strongly to $0$. We show, that each subsequence of $\{Tx_n\}$ has a convergent subsequence. So here, we can find a constant $C>0$ and $A$ infinite such that $\lVert x_n\rVert\geq C$ for all $n\in A$. Let $y_n:=\frac 1{\lVert x_n\rVert}x_n$ for $n\in A$. Then $\{y_n\}_{n\in A}$ converges weakly to $0$, as for $f\in (\ell^r)^*$ and $n\in A$, we have $$|f(y_n)|=|f(x_n)|\frac 1{\lVert y_n\rVert}\leq \frac{|f(x_n)|}C.$$
By definition of equivalence $\{u_n\}$ and $\{v_n\}$ are equivalent if for all sequence $\{a_n\}$ of scalars, $\sum_{n=1}^{+\infty}a_nu_n$ is convergent if and only if so is $\sum_{n=1}^{+\infty}a_nv_n$. So $\{x_n\}_{n\in A}$ is equivalent to $\{e_n\}_{n\in A}$ (not to the whole sequence). But it's enough to conclude, as we would have boundedness from an infinite subspace of $\ell^r$ to $\ell^p$.
I think that for your first question the answer is:
If $\|x_{n}\|\rightarrow 0$, then $x_{n}\rightarrow 0$ which implies that $T(x_n)\rightarrow 0$. So you can suppose that $\|x_{n}\|>\delta$ for some $\delta>0$. By hypothesis you have $$\langle y,x_{n}\rangle\rightarrow 0,\ \forall\ y\in X^{\star}$$
hence $$\Big|\Big\langle y,\frac{x_{n}}{\|x_{n}\|}\Big\rangle\Big|\leq \delta|\langle y,x_{n}\rangle|,\ \forall\ y\in X^{\star} $$
From the last inequality you can conclude that $\frac{x_{n}}{\|x_{n}\|}$ is an weakly null sequence.