what is the relationship between the blockchain size and the transaction volume on the network? I'm thinking about the scalability of the system with regard to the transaction volume. Is it possible to predict future blockchain size for a certain future transaction volume? Is the relationship directly proportional?

Are there any estimates compared to traditional fiat currencies transaction volumes?
Well, there are upper per-block size limits (about 500KB per block) based on the currently implemented protocol.  So max levels might be 5x or 10X current levels.

When those levels get hit, then it will be levels of fees that determine transaction volumes, and the many microtransactions seen today will no longer be justifiable.