I will try to answer the second question.
Let ${\bf C} \xleftarrow{i} {\cal A} \xrightarrow{f} {\bf B}$ be a span where $i$ is dense and fully faithful. Moreover $\text{lan}_if$ is pointwise. Then, the following are equivalent.
- $\text{lan}_if \dashv \text{lan}_fi$. 2. $f$ is the $i$-relative left adjoint of $\text{lan}_fi$, i.e. ${\bf C}(i, \text{lan}_fi) \cong {\bf B}(f, \_ ).$
If $i$ is only fully faithful $1 \Rightarrow 2$, if $i$ is only dense $2 \Rightarrow 1$.
Proof.
$1 \Rightarrow 2$) $${\bf B}(f, \_) \stackrel{i \text{ is ff.}}{\cong} {\bf B}((\text{lan}_if) i, \_) \stackrel{1}{\cong} {\bf C}(i, \text{lan}_fi).$$
$2 \Rightarrow 1$) $${\bf B}(\text{lan}_if, \_) \stackrel{\text{point.}}{\cong} \text{ran}_i{\bf B}(f, \_) \stackrel{2}{\cong} \text{ran}_i{\bf C}(i, \text{lan}_fi) \stackrel{\text{point.}}{\cong} {\bf C}(\text{lan}_ii, \text{lan}_fi) \stackrel{i \text{ is dense}}{\cong} {\bf C}(\_, \text{lan}_fi).$$