Data structures for embedded simplicial complexes
First you can read the CGAL documentation that can help you to understand combinatorial and generalized maps, since it provide several examples. You can also read the book "Combinatorial Maps: Efficient Data Structures for Computer Graphics and Image Processing".

Now some answers:

1) I don't think. In a combinatorial map, in 2D, a triangle is described by 3 darts. In 3D, a tetrahedron is described by 4 triangles, thus 12 darts. Etc...

Given a dart $d$, each permutation, except $\beta_n$, stay in the same n-simplex. A dart corresponds to a n+1-tuple (vertex, edge, face, ..., n-simplex), and each $\beta_i$ (except $\beta_1$) can be seen as a switch for the $i^{th}$ element of the tuple (each switch also switches vertices).

2) Condition 4 ensures that when two n-simplices are glued along one of their (n-1)-face, all the darts of the two (n-1)-faces are correctly identified two by two (cf. the following example).

3) Only quasi-manifold. For example in 3D, you cannot represent two tetrahedra that share only a common vertex, since this is not a quasi-manifold.

4)  A dD quasi-manifold is an object obtained by taking some isolated d-cells, and allowing to glue d-cells along (d-1)-cells (cf. this section).

5) Let us consider one simple example in 2D. Take a square with four edges a,b,c,d. Now you can identify edges a and c to form either an orientable strip or a Mobius strip. Thus you need two different type of identification. This is not possible with combinatorial maps but this is possible with generalized maps since one edge in 2D is described by two darts, allowing to choose which pair of darts are identified (cf. the example here).

Hope this helps, and welcome into the fabulous world of combinatorial maps !