What thin metals in sheets can I solder?

Brass sheet and small diameter rod is commonly available. Not cheap and relatively heavy for its strength/stiffness.

As an alternative to soldering, have a look at the foam/fibreglass materials used for constructing model aircraft. Excellent strength to weight, easy to cut/join using 5 minute epoxy. For instant repairs you can even use hot melt glue.

0.4mm fibreglass/epoxy sheet can be cut with scissors. The 1.6mm thick fibreglass PCB material is super strong but requires a diamond wheel/Dremel tool and dust extraction.

Bamboo barbecue skewers from your local supermarket are cheap, light, super strong and easy to glue.