1. Using the answer to (7), if the natural transformation $J \Rightarrow J'$ is invertible, and $G$ and $F'$ are fully faithful, then the pasting square will commute up to natural isomorphism (assuming $J$ is dense and fully faithful).

2. Again using the answer to (7), if $F$ and $G_1$ are fully faithful, then the composite natural transformation will be invertible (assuming $J$ is dense and fully faithful). As far as I can tell, the second part of the question is ill-formed.

3. Yes, $F$ is a $J$-relative left-adjoint to $G$ if and only if $f \cong \mathrm{lift}_G J$ and this left lifting is absolute (this is an easy exercise). However, this is not true for enriched relative adjunctions. An example of non-unique relative right adjoints is given in Ulmer's *Properties of Dense and Relative Adjoint Functors*: let $J : \mathrm{AbGrp}_f \to \mathrm{AbGrp}$ be the inclusion of finite Abelian groups in all Abelian groups and let $L : \mathrm{AbGrp}_f \to \mathrm{Vect}_{\mathbb Q}$ be the zero functor. Then $L$ is $J$-relative left adjoint both to $0 : \mathrm{Vect}_{\mathbb Q} \to \mathrm{AbGrp}$ and to the forgetful functor $V : \mathrm{Vect}_{\mathbb Q} \to \mathrm{AbGrp}$. I'm not aware of any nice structure on the set of $J$-relative right adjoints to a given functor.

4. Adjunctions relative to isomorphisms are the same as adjunctions relative to the identity. There is no lack of uniqueness here, since isomorphisms are dense, and $J$-relative right adjoints are unique when $J$ is dense.

5. Weighted colimits as relative adjunctions are treated in §4 of Street–Walter's *Yoneda structures on 2-categories*, where they are called "weak indexed colimits". See there for various useful lemmas making use of this characterisation.

6. If the identity on $\mathbf A$ is $J$-relative left coadjoint to $G : \mathbf B \to \mathbf A$, then there is a natural isomorphism $\mathbf A(a, Jb) \cong \mathbf A(a, Gb)$ which, by Yoneda, just says that $J \cong G$.

7. Assume $J$ is dense and fully faithful. If $L$ is $J$-relative left adjoint to $R$, then the unit $\eta : J \Rightarrow RL$ is an isomorphism if and only if $L$ is fully faithful (easy exercise). Dually, assume $J$ is codense and fully faithful. If $L$ is $J$-relative left coadjoint to $R$, then the counit $\epsilon : LR \Rightarrow J$ is an isomorphism if and only if $R$ is fully faithful.