$p = \sqrt{1+\sqrt{1+\sqrt{1 + \cdots}}}$; $\sum_{k=2}^{\infty}{\frac{\lfloor p^k \rceil}{2^k}} = ? $

You are correct that $$p=\frac{1+\sqrt{5}}{2}$$

This is the golden ratio, ususally labeled $\phi$. There is a well-known relationship between $\phi$ and the Lucas numbers $L_k$: for all natural numbers $k$,

$$L_k = \phi^k + \left(\frac{-1}{\phi}\right)^k$$

The second term above will be less than $1/2$ for $k >1$, so rounding $\phi^k$ to the nearest integer must give $L_k$.

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This means your sum can be written in terms of Lucas numbers. Let your sum be $S$. Then

$$S = \displaystyle\sum\limits_{k=2}^{\infty} \frac{L_k}{2^k}$$

Now we can write out a few terms of $S$, $2S$, and $4S$:

\begin{align} S = \frac{L_2}{4}&+\frac{L_3}{8}+\frac{L_4}{16}+\frac{L_5}{32}+\cdots \\\\ 2S = \frac{L_2}{2}+\frac{L_3}{4}&+\frac{L_4}{8}+\frac{L_5}{16}+\cdots \\\\ 4S = L_2+\frac{L_3}{2}+\frac{L_4}{4}&+\frac{L_5}{8}+\frac{L_6}{16}\cdots \\ \end{align}

I've intentionally lined up the sums above like that because if we add up the first two equations, term by term, we get

$$ 3S = \frac{L_2}{2} + \frac{L_2+L_3}{4} + \frac{L_3+L_4}{8}+\frac{L_4 + L_5}{16}+\cdots $$

But the Lucas numbers, like the Fibonacci numbers, satisfy the recursion $L_k + L_{k+1} = L_{k+2}$. Therefore

$$3S = \frac{L_2}{2} + \left(\frac{L_4}{4} + \frac{L_5}{8} + \frac{L_6}{16}+\cdots\right)$$

But everything after the first term looks a lot like what we had written for $4S$ above! In fact, we can write

$$3S = \frac{L_2}{2} + \left(4S - L_2 - \frac{L_3}{2}\right)$$

$$S = \frac{L_2}{2} + \frac{L_3}{2}$$

$$S = \frac{L_4}{2}$$

Finally, we can solve for your sum:

$$S = \frac{7}{2}$$

For $n \ge 2$, the numbers $\lfloor \phi^k \rceil$ are equal to the Lucas numbers, which satisfy the Fibonacci recurrence but in which $L_1=1$ and $L_2=3$. The Lucas numbers satisfy

$$L_k = \phi^k + (-1)^k \phi^{-k}$$

and thus have the generating function

$$L(x) = \frac{2-x}{1-x-x^2} $$

so the sum we want is

$$L \left (\frac12 \right ) - L_0 - \frac12 L_1 = \frac72$$

Note that $\phi^{-1}= \phi-1 = 0.618\ldots$ and exponentiation to the power of $k$ does not affect a rounding for $k \ge 2$.