What is the free lax-idempotent adjunction?

Question

Let $Adj$ be the free adjunction, i.e. the 2-category such that for any 2-category $K$, the functor 2-category $2Fun(Adj, K)$ is the 2-category of adjunctions in $K$ (naturally in $K$). Note that $Adj$ is a finitely-presentable 2-category: it is generated by 4 cells (the left adjoint, the right adjoint, the unit, and the counit) subject to 2 equations (the triangle equations). And the free monad $Mnd = B(\Delta_{aug})$ (the delooping of the augmented simplex category $\Delta_{aug}$, which is the free monoidal category on a monoid object, or equivalently a certain full sub 1-category of $Cat$, endowed with a monoidal structure) embeds as a full subcategory of $Adj$. There is also a description of the 4 hom-categories of $Adj$ as certain non-full subcategories of $\Delta_{aug}$.

Now let $Mnd^\ell = B(\Delta_{aug}^\ell)$ be the free 3-category on a lax-idempotent (a.k.a. Kock-Zoberlein) 2-monad. Here $\Delta_{aug}^\ell$ is a certain full sub 2-category of $Cat$ -- its 1-skeleton is $\Delta_{aug}$ but it also has some 2-cells exhibiting the elementary face / degeneracy maps as occurring in adjoint section/retract strings. Note that it is a property for a monad to extend to a lax-idempotent monad.

Question A: Does there exist a 3-category $Adj^\ell$ with the following properties?

1. It is a property for an adjunction $Adj \to K$ to extend to a "lax-idempotent adjunction" $Adj^\ell \to K$.

2. An adjunction $Adj \to K$ extends to $Adj^\ell \to K$ if and only if the restriction to a monad $Mnd \to K$ is lax-idempotent, extending to $Mnd^\ell \to K$, if and only if the restriction to a comonad $Mnd^{co} \to K$ is (co?)lax-idempotent, extending to $(Mnd^\ell)^{co} \to K$.

I think the answer is "yes", and assuming this I have some follow-up questions:

Question B: Does $Adj^\ell$ coincide with the "obvious" 3-category whose hom 2-categories are certain locally-full sub-2-categories of $\Delta_{aug}^\ell$?

Question C: Is $Adj^\ell$ finitely-presentable as a 3-category? (For that matter, I would assume that $Mnd^\ell$, like $Mnd$, is not finitely-presentable as a 3-category -- is that correct?) For example, is $Adj^\ell$ obtained from $Adj$ by forcing the unit and / or counit to have the appropriate co/retract adjoint?

Question D: Does any of this depend on whether we work weakly or strictly?