lower bound for Renyi–Ulam Game with lies
For the first question, suppose that you have an algorithm that supports one error, and uses $m$ questions. For each element $i \in \{1,\ldots,n\}$, let $x_{i0}$ be the vector containing the answers to all questions when A doesn't lie, and let $x_{ij}$ be the vector containing the answers to all questions when A lies in question $j$. In total, we have $n$ collections of $m+1$ vectors. All these vectors are distinct: the $m+1$ vectors in a collection are distinct by definition, and vectors belonging to different collections must be distinct since the correct answer is different. We deduce that $(m+1)n \leq 2^m$, and so $m \geq \log n + \log\log n + \Omega(1)$.

For the second question, here is something to try. Let $m$ be a number so that $\log n$ queries suffice to find an element in $\{1,\ldots,m\}$, even when A is allowed to lie at most once. Partition $\{1,\ldots,n\}$ into $m$ intervals $[x_1,x_2),[x_2,x_3),\ldots,[x_m,x_{m+1})$, where $[a,b) = \{a,\ldots,b-1\}$, $x_1 = 1$, and $x_{m+1} = n+1$. Then this gives an $\alpha$-approximation for $\alpha = \max \frac{x_{i+1}-1}{x_i}$. I leave the rest to you.

The third question is addressed in Three thresholds for a liar by Spencer and Winkler. The threshold is $n/3$ (a strategy exists if $\ell &lt; n/3$).