On optimality of Grover algorithm with high success probability
For the sake of completeness, here's an answer. 

Let $Q_\epsilon(f)$ denote the $\epsilon$-error quantum query complexity of computing a function $f$ and $\mathsf{OR}_n$ be the OR function on $n$ bits, defined as $\mathsf{OR}_n(x_1,\ldots,x_n)=\bigvee_{i=1}^n x_i$. (Note that this is different from the problem where you are promised that the input has exactly one 1 and the objective is to find that 1. That problem can be solved with no error in $\Theta(\sqrt{n})$ queries.)

Then we have for all $\epsilon \in [2^{-n},1/3]$,


  $Q_\epsilon(\mathsf{OR}_n) = \Theta(\sqrt{n\log(1/\epsilon)})$.


This follows from Bounds for Small-Error and Zero-Error Quantum Algorithms.

In fact, we know something more general. For all symmetric functions $f$, which are functions that only depend on the Hamming weight of the input, we have that for all $\epsilon \in [2^{-n},1/3]$,


  $Q_\epsilon(f) = \Theta(Q_{1/3}(f)+\sqrt{n\log(1/\epsilon)})$.


This was shown in A note on quantum algorithms and the minimal degree of epsilon-error polynomials for symmetric functions.