For your first question, that is somehow the point with KZ-monads: they don't look idempotent (in the definition) but they do behave a little bit like idempotent monad in practice: for a KZ-monad, an object can only have one structure of algebra, but it is not true that $TT X$ is the same as $ TX$ and not every morphism between the underlying objects is a morphisms of algebra. But every morphism between the underlying object extend in a unique way as a lax morphisms of algebra.
While for a really idempotent monad, every morphism of the underlying object is a morphism of algebra.

The typical example of KZ-monad is the free co-completion monad (let say under finite colimits to avoid size problems, but they are not very important):

An algebra is a finitely co-complete category, a morphism of algebra a finite co-limit preserving functor. But the the co-completion of a co-complete category (seen as a category) is not the category itself (hence there is not real "idempotency" of the monad").


First this definition in terms of finite ordinal is exactly A.Kock's definition in ['Monads for which structures are adjoint to units'][1] (other [link][2] to the paper) coupled to A.Kock's description of the monoidal 2-category $\Delta$ en term of generator and relations given in [Generators and relations for $\Delta$ as a monoidal $2$-category][3].


A way to explain the relation to idempotency directly on the definition, is that you can say that an idempotent monad is exactly a mondad $T$ where the two natural map $T(X) \rightrightarrows TT(X)$ are equal. 
A 'pseudo-idempotent' monad (on a 2-category) would be a monad where there is an isomorphism between these two $1$-cell $T(X) \rightrightarrows TT(X)$ satisfying some coherence condition. A lax-idempotent monad (or KZ-monad) is when you just have a non-invertible $2$-cell between these two $1$-cell also satisfying some coherence conditions.

Only pseudo-idempotent monad would really be "2-dimensional generalization of idempotent monad" in the sens that they would satisfies that $TT(X)$ is isomorphic to $T(X)$ for all $X$.

For more detail, have a look to A.Kock's paper linked above where he develope the theory of such monad (he calls them KZ-doctrines)

  [1]: http://home.math.au.dk/kock/msau.PDF
  [2]: https://www.sciencedirect.com/science/article/pii/002240499400111U
  [3]: http://citeseerx.ist.psu.edu/viewdoc/download;jsessionid=DD11122A669330B4F3B634C0305E71F0?doi=10.1.1.38.1851&rep=rep1&type=pdf