Confusion around the reflection equation algebra

1. The only reasonable definition of the REA associated with a quasi-triangular Hopf algebra is 1). This is, of course, a somewhat abstract definition but provides a solution of the RE which is universal in a precise sense.
2. is reminiscent of the so-called Faddeev-Reshetikhin-Takhtajan (usually abbreviated as FRT) construction. Its main advantage is that it does not require an Hopf algebra to start with (rather, in the original FRT construction, the goal was to produce a Hopf algebra starting from an arbitrary solution of the QYBE). EVen if $R$ do come from a quasi-triangular Hopf algebra, it won't give the same answer as 1) except in the case of $U_q(\mathfrak{gl}_n)$ (even then this is not quite true, you get some deformation of $\mathcal O(\mathfrak{gl}_n)$ rather than $\mathcal O(GL_n)$). In general there will be a map from 2) to 1).
3. On the other hand as you say, you can run this construction in the case $V$ is some representation that generates all the other. Indeed this approach is useful to find a presentation of the REA, since it is indeed generated by matrix coefficients: roughly speaking this will give you a set of generators but not in general all the relations. This is what happens here: if you run the FRT-like reconstruction for the R-matrix of $\mathfrak{sl}_n$ you get some algebra, but then you need to add this extra relation you mention which, as you probably know, is nothing but a $q$-analog of $\det(A)=1$. Again this already shows up in the original situation, see Definition 4 in http://math.soimeme.org/~arunram/Resources/Reshetikhin/QuantizationOfLieGroupsAndLieAlgebras.html.

Edit It's useful to think about universal properties: 1) is universal for algebras $A$ with a solution of the RE in $A\otimes H$, while 2) is universal for algebras $A$ with a solution of the RE in $A\otimes End(V)$. Of course, composing with the algebra map $H\rightarrow End(V)$ given by the action of $H$ on $V$ every solution of the first equation gives you a solution to the second one, so applying this to the case $A$ is the REA itself you get a map from the algebra constructed in 2) to the one constructed in 1).

Let me first note that the reflection matrix, which you denote by $A$, is often called K-matrix, cf its graphical representation with | a 'wall' and < the 'worldline' of particle bouncing off the wall. The graphical form of the equation can already be found in Cherednik, Factorizing particles on a half-line and root systems (1984) https://link.springer.com/article/10.1007/BF01038545. The notation $K$ might be due to Sklyanin, Boundary conditions for integrable quantum systems (1988), https://iopscience.iop.org/article/10.1088/0305-4470/21/10/015.

The reflection (equation) algebra is the reflection-equation analogue of the Yang--Baxter algebra: to any choice of finite-dimensional vector space and R-matrix obeying the Yang--Baxter equation (and suitable other properties, such as braiding unitarity and an 'initial condition') one can associate a unital associative algebra generated by the operator-valued (noncommutative) entries of the K-matrix obeying the reflection equation.

If one would replace the reflection (`$RKRK$') equation by the $RLL$-equation one instead arrives at the Yang--Baxter algebra, which is the operator algebra closely related to the FRT (or R-matrix) presentation of quantum affine algebras.

Re 3: The FRT presentation says nothing about the quantum determinant, so to get $SL_n$ you need to impose $qdet = 1$ separately, which is your last equation in 3. The version of the reflection equation that you give there can sometimes be simplified: Suppose that the R-matrix is symmetric in the sense that $P R P = R$ with $P$ the permutation. Then $R_{21} = R_{12}$ in the usual tensor-leg notation. In such cases all R-matrices in the reflection equation can be written using just $R$. (Graphically the need for $R_{21}$ is clear, though.)

Re 2: These authors work with the braid-like version of the R-matrix, often denoted by $\check{R}$. Namely, suppose that $R$ obeys the YBE

$$ R_{12}(u,v) \ R_{13}(u,w) \ R_{23}(v,w) = R_{23}(v,w) \ R_{13}(u,w) \ R_{12}(u,v) \ , $$

where I have assumed that the R-matrix might depend on a spectral parameter associated to each copy of the auxiliary space in general. (This is for the affine case, but helps highlighting the structure of the equation.) Then both of $P \ R$ and $R \ P$ obey the braid-like version of the YBE

$$ \check{R}_{12}(u,v) \ \check{R}_{23}(u,w) \ \check{R}_{12}(v,w) = \check{R}_{23}(v,w) \ \check{R}_{12}(u,w) \ \check{R}_{23}(u,v) \ . $$

You always have to check which version is used. In the paper you cite in 2 it's the latter, which is why both $A$s have the same subscript.

Re 1: I believe that the proper algebraic interpretation of Sklyanin's construction of representations of the K-matrix as the double-row monodromy matrix, constructed from a K-matrix with scalar entries and an L-operator, is as a coideal subalgebra, see Kolb and Stokman, Reflection equation algebras, coideal subalgebras, and their centres, https://arxiv.org/abs/0812.4459.

You might also be interested in the recent paper by Appel and Vlaar, Universal k-matrices for quantum Kac-Moody algebras, https://arxiv.org/abs/2007.09218