A question on staking, gas fees, and layer 2 rollups

I understand Ethereum’s roadmap of PoS transition, followed by data sharding. This will empower rollups to function as the execution layer over the underlying Ethereum PoS consensus layer, ensuring massive scalability while inheriting Ethereum’s security and decentralisation.

I have two questions regarding the future of the system:

1. How will it be feasible to move ETH around for the purposes of staking? This seems like the final, fundamental use for L1 outside of rollup batches. If gas fees on L1 are likely to continue increasing as adoption grows, how will new validators establish themselves when they need to pay high gas fees? Will there be staking functionality somehow offered through L2, essentially batched as well? How would this affect security?

2. Using optimistic or ZK proofs, rollups can securely handle execution while outsourcing to L1 for consensus. However, while proofs can ensure that transactions batched to L1 are valid, what prevents rollups censoring certain transactions before they are committed to L1?

Thanks!

Edit: I’ve greatly appreciated posts by u/Liberosist so perhaps they might have some insight here.