The cages weight certainly does matter as it is taken into account when you have to factor in the stress that it can support.
Building (or rather, DESIGNING) a cage is much more complicated than it seems. Your initial images did not show up for me, so I'm not entirely sure what you are planning to do, but if you would like some help/questions on the design feel free to shoot me an IM/PM and I can help you out.
You can build something super-simplistic and the design will be easy...but the more things you incorporate, the complexity adds up fast...
@Eltortu's response: I have to disagree with him on this. I have used standard pine 2x4's and plyboard for several builds now and it is holding up perfectly fine (and is far cheaper than any alternative wood that I could find). I believe the secret to this is to
PROPERLY SEAL YOUR WOOD. I use an All-Purpose Weatherproof Deck Stain by Behr designed to withstand hail and snowstorm conditions. My oldest build with this technique is now about 7.5 months old misted daily with no bending or any other signs of water damage. I do apply at least 1 coat to each piece of wood BEFORE connecting it and then 1 final coat after the structure is complete.
If you do not properly seal your wood and/or use a wood which does not react well to water....you will be looking at both mold/fungus issues and potential structural issues down the line. (This is also another factor where weight does play into your design....IF it does weaken, you need make sure it doesn't buckle)
TO CLARIFY*** When I said I use 2x4's I don't mean for the entire thing. I use 2x4's for my structural studs/supports combined with 1/2"-3/4" plyboard (depending on structural needs) for the main body. If you are planning to plank 2x4's...that might be pretty heavy....but if you are just using 2x4's for a skeletal frame and running screen over everything else (all 5 sides) then I can say with confidence that structurally it will hold fine....though it will still be kinda heavy (but I actually find that to be a benefit, as it makes it more sturdy as well....you won't move it THAT much)
You can AIM me @ ManiArabi and i'll be glad to take a look at your specs
EDIT: Depending on the dimensions of your overall cage, 2x4's might be too big....you can prob get away with 1x4's or maybe even 1x2's for a smaller cage (like 24"x24"x48" or so) depending on the other weight factors you are putting in