Can any node on tezos L1 refute a rollup proof? Will the new octez for L1 nodes for Mumbai implements something that automatically checks if the rollups proofs published are valid, and if not stake 5k tez?
Anyone can refute the proof. I doubt that auto-refute will be available out-of-the-box but I'm sure that this "liquidating bot" for rollups will come out if it's necessary. I feel like most refutations will come from software errors rather than pure malice. Same with double baking.
Yes and no. When someone challenges a state commitment posted by a rollup operator, the L1 protocol adminstrates that challenge and slashes the loser. Because it's enshrined in the L1 protocol, all bakers help to secure this process. However, the octez L1 node only checks these fraud proofs, and does not compute any state commitments. That's where the scalability comes from - you can run a rollup node and validate only the rollup you care about.
First, to be clear: With optimistic rollups, proofs are only posted in case of a dispute (unlike zk/validity-rollups where a proof is posted with every commitment). Such a dispute will be between the rollup node that posted a given commitment representing the state of the rollup, and another rollup node disputing the validity of the commitment by publishing a concurrent commitment.
These are the parties playing out the refutation game and are staking tez in that context. Anyone can run a dedicated rollup node and dispute a commitment and enter into a refutation game, but it’s not the default role of Layer 1 nodes.
The role of Tezos' Layer 1 nodes is to decide who wins the so-called refutation game, and yes, in that case all Layer 1 nodes participate, but without a stake in the game. It acts as an arbiter for the game.
A rollup is safe as long as there is at least one honest participant: if someone is interested in checking what is happening in a given rollup, she can run a rollup node in accuser mode: if a dishonest commitment is published on the Layer 1, the rollup node will publish the honest commitment, bond the 10k tez required, and play the refutation game. If the participant is honest, the participant will win the refutation game and will be rewarded with 5k tez.