They've hit 8 tx/sec in production and 25 tx/sec on the testnet. Proof of stake should increase capacity to over 100 tx/sec, and a simple sharding scheme to at least 10,000 tx/sec. Long-term they hope to do even better.

Blockchain size is increasing rapidly, I think it's up to 40GB already. But they plan to implement pruning before it becomes a serious problem.