Interesting - I've been thinking along similar lines recently, especially the stream of transactions. I was leaning more towards having each node 'sign' a transaction with it's stake as they get relayed between nodes, such that you get a fast forming, POS style consensus on each transaction as it passes between nodes. The only (and significant) problem with this is that it causes transactions to grow as they pass between nodes.
Hmm, depends if this growth is ephemeral or not. If it's cleared away somehow, PoW style, it could be tenable, though there might be trouble making the PoS mean anything to the miner; it would may necessitate everyone synchronising their mempools, to some degree at least, or including the PoS stuff in the block, though that would defeat the purpose. Hmm, indeed.