I will try to answer the second question. 

>**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.

>1. $\text{lan}_if \dashv \text{lan}_fi$.
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.

$1 \Rightarrow 2$)  $${\bf B}(f, \_) \stackrel{i \text{ is ff.}}{\cong} {\bf B}((\text{lan}_if) i, \_) \stackrel{1}{\cong} {\bf C}(i, \text{lan}_fi).$$

$2 \Rightarrow 1$)
$${\bf B}(\text{lan}_if, \_) \stackrel{\text{point.}}{\cong} \text{lan}_i{\bf B}(f, \_) \stackrel{2}{\cong} \text{lan}_i{\bf C}(i, \text{lan}_fi) \stackrel{\text{point.}}{\cong} {\bf C}(\text{lan}_ii, \text{lan}_fi)  \stackrel{i \text{ is dense}}{\cong} {\bf C}(\_, \text{lan}_fi).$$

$3$ is just a rewriting of $2$.


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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.  ${\bf A}$ is small, ${\bf C}$ and ${\bf B}$ are cocomplete. Then, the following are equivalent.

>1. $\text{lan}_if \dashv \text{lan}_fi$.
>2. V preserves $\text{lan}_fi$.
>3. ${\bf C}(i,\text{lan}_fi) \cong  \text{lan}_f{\bf C}(i,i)$.


Proof.
Recall that $\text{lan}_if$ is pointwise,because ${\bf B}$ is cocomplete.

$1 \Rightarrow 2)$.

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 $2$.

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>**Cor. 1** (Quite surprising).
 Let ${\bf Set}^{A^\circ} \xleftarrow{y} {\bf A} \xrightarrow{f} {\bf B}$ be a span where $y$ is the Yoneda embedding of a small category and ${\bf B}$ is cocomplete. Then $\text{lan}_yf \dashv \text{lan}_fy$.

Proof 1. In Prop 1 the condition 2 is verified and is essentially a rewriting of the Yoneda Lemma. Moreover $\text{lan}_yf$ is pointwise because ${\bf B}$ is cocomplete.

Proof 2. In Prop 2 the condition 2 is trivially verified because $V$ is the identity.