Given the same thickness core of bamboo and paulownia, which is lightest, and which has more flex?
Also, is balsa suitable as a core material, or is it just not strong enough.

Both Bamboo and Paulownia come in various densities depending on how they are laminated and which part of the wood is used. You can make balsa cored boards but it's more suited to designs with higher skin integrity, ie all carbon. BEST made Balsa cored wakeboard style boards for 3 seasons, back in 04-07

this is how is see it.

balsa lightest
paulownia light and somewhat flexi
bamboo heaviest, strongest and most flexible.

I am using bamboo at the moment. I made 6 bamboo boards land year.
3 x mini longboard skateboards vlam and ply bamboo + triaxle glass
1 x minilongboard skateboard ply bamboo, foam core with carbon
1 x lanboard vlam bamboo, carbon.
1 x mutant kiteboard vlam bamboo triax glass and carbon stringer.

If you like flex and want a bomb proof board then bamboo is the way. It you want light and stiff then the other cores are going to give you a weight advantage.

Also the flex properties are different with bamboo too. I certainly love the feel of my mutant. Its the smoothest board i have ever ridden in the chop.