"Tomato cage" plastic connections -- Design idea
Typically a wire product like this would be electro-welded together. 



Your set of criteria is going to be next to impossible to achieve. You envision something that has the properties of a fully welded structure but desire it to be a pile of loose parts. 

I believe that your best goal would be to make each level of your gadget an electro-welded assembly that is self sufficient from a structural standpoint. The design should be such that these pieces are able to stack together. At each interface between the levels there would be adjacent horizontal rings that are part of the lower stacked element and the next stacked unit. You could then have the user/installer then tie these rings together using nylon zip-ties of an appropriate size. Dis-assembly is as simple as cutting these ties with a diagonal cutter. 

Cross sectional detail of one side. White dots are electro welds.