For 2. 

Claim: $\lim_{n\rightarrow\infty}2^{-n/2}\mathbb{E}[K_n]=0$

The idea is this: The union of the events $E_{jn}$ is the event 
\begin{equation}
F_n=\left\{B_t=0 \text{ for some $t\in\left[\frac{2^n+1}{2^n},\frac{2^{2n}}{2^n}\right]$} \right\}
\end{equation} with probability less or equal than 1. The intersection of events $E_{jn}$ and $E_{j+1,n}$ is
\begin{equation}
G_{jn}=\left\{B_{j/2^n}=0\right\}
\end{equation} which has probability equal to zero for all $j,n$. Then
\begin{eqnarray}
\mathbb{E}[K_n] & = & \mathbb{E}\left[\sum_{j=2^n+1}^{2^{2n}}1_{E_{jn}}\right]\\
& = & \sum_{j=2^n+1}^{2^{2n}}\mathbb{E}[1_{E_{jn}}]\\
& = & \sum_{j=2^n+1}^{2^{2n}}\mathbb{P}[E_{jn}]\\
& = & \mathbb{P}[F_{n}]+\sum_{j=2^n+1}^{2^{2n}}\mathbb{P}[G_{jn}]\\
& \leq & 1
\end{eqnarray}

This implies $\lim_{n\rightarrow\infty}2^{-n/2}\mathbb{E}[K_n]=0$