1. You can check that the distances and angles are the same between the 2 sets. If not, then they aren't equivalent under translations and rotations. This is a necessary condition, not a sufficient one.

2. You can get some information out by working with (f-e) = R (b-a). By choosing different pairs of e,f,g,h, you should be able to constrain R. Once you have R, getting T should be relatively easy.