I will try to answer the second question. Notice that the main results of this answer are Cor 1 and Cor 2, but Prop 1 and 2 have a theoretical interest.
Cor 1. Let ${\bf C} \xleftarrow{i} {\bf A} \xrightarrow{f} {\bf B}$ be a span where $i$ is dense and fully faithful. Moreover ${\bf A}$ is small, ${\bf C}$ and ${\bf B}$ are locally presentable. Then $\text{lan}_if \dashv \text{lan}_fi$.
Cor 2. Let ${\bf C} \xleftarrow{i} {\bf A} \xrightarrow{f} {\bf B}$ be a span where $i$ is dense and fully faithful. Moreover ${\bf A}$ is accessible, ${\bf C}$ and ${\bf B}$ are locally presentable, $i$ and $f$ are accessible. Then $\text{lan}_if \dashv \text{lan}_fi$.
Prop 1. Let ${\bf C} \xleftarrow{i} {\bf A} \xrightarrow{f} {\bf B}$ 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. ${\bf C}(i, \text{lan}_fi) \cong {\bf B}(f, \_ ).$ 3. $f = \text{lift}_{\text{lan}_fi}i$ and the lift is absolute.
If $i$ is only fully faithful $1 \Rightarrow 2$, if $i$ is only dense $2 \Rightarrow 1$.
Proof.
$1 \Rightarrow 2$) $${\bf B}(f, \_) \stackrel{i \text{ is ff.}}{\cong} {\bf B}((\text{lan}_if) i, \_) \stackrel{1}{\cong} {\bf C}(i, \text{lan}_fi).$$
$2 \Rightarrow 1$) $${\bf B}(\text{lan}_if, \_) \stackrel{\text{point.}}{\cong} \text{ran}_i{\bf B}(f, \_) \stackrel{2}{\cong} \text{ran}_i{\bf C}(i, \text{lan}_fi) \stackrel{\text{point.}}{\cong} {\bf C}(\text{lan}_ii, \text{lan}_fi) \stackrel{i \text{ is dense}}{\cong} {\bf C}(\_, \text{lan}_fi).$$
$3$ is just a rewriting of $2$.
Now we study a very special setting.
Let ${\bf C} \xleftarrow{i} {\bf A} \xrightarrow{f} {\bf B}$ be a span where $i$ is dense and fully faithful. Moreover $\text{lan}_if$ is pointwise, ${\bf A}$ is small, ${\bf C}$ and ${\bf B}$ are cocomplete.
In this setting ${\bf C}$ is a reflective subcategory $ V: {\bf C} \leftrightarrows \text{Set}^{{\bf A}^\circ} : L $ of $\text{Set}^{{\bf A}^\circ}$ via the nerve $V = \text{lan}_i(y_{{\bf A}})$ (V is the right adjoint).
Prop 2. Let ${\bf C} \xleftarrow{i} {\bf A} \xrightarrow{f} {\bf B}$ be a span where $i$ is dense and fully faithful. Moreover $\text{lan}_if$ is pointwise, ${\bf A}$ is small, ${\bf C}$ and ${\bf B}$ are cocomplete. Then, the following are equivalent.
- $\text{lan}_if \dashv \text{lan}_fi$.
- V preserves $\text{lan}_fi$.
- ${\bf C}(i,\text{lan}_fi) \cong \text{lan}_f{\bf C}(i,i)$.
Proof.
$1 \Rightarrow 2)$.
Using Prop 1. we know that ${\bf C}(i, \text{lan}_fi) \cong {\bf B}(f, \_ )$. Since the presheaf construction is a Yoneda structure, we have that $\text{Set}^{{\bf A}^\circ}(y_A, \text{lan}_fy_A) \cong {\bf B}(f, \_)$.
Thus, $$ {\bf C}(i, \text{lan}_fi) \cong {\bf B}(f, \_) \cong \text{Set}^{{\bf A}^\circ}(y_A, \text{lan}_fy_A) \cong \text{Set}^{{\bf A}^\circ}(y_A, \text{lan}_fVi)$$
Observe also that $${\bf C}(i, \text{lan}_fi) \cong {\bf C}(Ly_A, \text{lan}_fi) \stackrel{L \dashv V}{\cong} \text{Set}^{{\bf A}^\circ}(y_A, V\text{lan}_fi),$$ putting the last two equation together, one gets: $$\text{Set}^{{\bf A}^\circ}(y_A, \text{lan}_fVi) \cong \text{Set}^{{\bf A}^\circ}(y_A, V\text{lan}_fi). $$ By Yoneda Lemma the two functors on the right have to coincide.
$2 \Rightarrow 1).$
Using Prop 1. it is enough to prove that ${\bf C}(i, \text{lan}_fi) \cong {\bf B}(f, \_ )$. Since the presheaf construction is a Yoneda structure, we have that $\text{Set}^{{\bf A}^\circ}(y_A, \text{lan}_fy_A) \cong {\bf B}(f, \_)$. Now, $${\bf C}(i, \text{lan}_fi) \cong {\bf C}(Ly_A, \text{lan}_fi) \stackrel{L \dashv V}{\cong} \text{Set}^{{\bf A}^\circ}(y_A, V\text{lan}_fi) \stackrel{2}{\cong} \text{Set}^{{\bf A}^\circ}(y_A, \text{lan}_fVi) \cong \text{Set}^{{\bf A}^\circ}(y_A, \text{lan}_fy_A) \cong {\bf B}(f, \_). $$
$3$ is just a rewriting of $2$.
Cor 1. Let ${\bf C} \xleftarrow{i} {\bf A} \xrightarrow{f} {\bf B}$ be a span where $i$ is dense and fully faithful. Moreover ${\bf A}$ is small, ${\bf C}$ and ${\bf B}$ are locally presentable. Then $\text{lan}_if \dashv \text{lan}_fi$.
Proof.
If needed, I will include a proof for this corollary. It is based on checking that the condition $3$ of Prop 2. is always verified.
Cor 2. Let ${\bf C} \xleftarrow{i} {\bf A} \xrightarrow{f} {\bf B}$ be a span where $i$ is dense and fully faithful. Moreover ${\bf A}$ is accessible, ${\bf C}$ and ${\bf B}$ are locally presentable, $i$ and $f$ are accessible. Then $\text{lan}_if \dashv \text{lan}_fi$.
Proof.
The proof is a bit long, I will write it if needed, it is based on finding a way to apply Cor 1. to this situation.