You are talking about this, a solderless breadboard ?
It does have power rails.
For crocodile clips (or multimeter), simply insert single header pins inverted into the breadboard at an appropriate point.
Multiple breadboards can be fixed together using the built mounting lugs.
I can't imagine a 3 dimensional construction using breadboards. The components may fall out. The same with turning the board over to use the proposed measurement points on the back.
The main problem I have with breadboards is the reliability of the connections, not its fundamental design. Maybe that is where you should focus your efforts in the contact design or the contact plating material.