How did the transaction signature verification process work in ETH before docking and how does it work today in ETH 2.0?

Hi, is anyone able to help me figure out how and at what stage signature verification takes place in the ETH network. Let's break this question into 2 parts.

1. regarding the original version of ETH (layer one before the docking/merge). Were miners involved in signature verification before entering transactions into the block and before guessing the nonce?

2. Regarding ETH 2.0. Who selects the transactions that are then to be added to the block? Is it the block proposer, the validators, or is there a smart contract that does this? (I am referring to situations without rollups).

Let's assume that the transactions are already selected. Is the next step the verification of signatures by each validator? I know that in the case of rollups this can be done off-chain, however without them at some stage the transactions must be verified, am I right?

Please, is anyone able to help me understand this?