In order to change the in-game attires to other unused ones, you can either redump the textures with new attire parts or you can rename the textures that you want with the names of the textures that are already used in the game. You can find tutorials on these in the tutorial section.
Additional arenas can be selected by pressing the C-Directional buttons while you are on the arena select screen.
The entrance music that you saw in the video was done using JamStubbs’ NMThing Music Tool. You can find a tutorial on how to use it here:
http://s4.zetaboards.com/Old_Skool_Reunion?topic=8145798/1/
The finishers that don't need grapples is not really a feature of Showdown 64. However, I along with the other contributors to this project are now working on a new mod named AKI Evolution, which may have features similar to this in the future. You can find information on it here:
http://s4.zetaboards.com/Old_Skool_Reunion?topic=30167855/1