Do I lose my ETH when doing a L2 Transaction over Optimism - NEED HELP!

Hi, I've already read many articles about Layer 2 Chains like Optimism, Arbitrum etc.

But there is one Question about the behavior of L2 Chains which didn't got answered properly so far.

Idea/Question:

Let's say I bought 1ETH on a Crypto Exchange like Bitget, which offers L2 Transactions of Ether.

And I now would send this ETH over the L1 Mainnet, it would go from Wallet A(Bitget) to Wallet B(Ledger) with ease and without any risk, but also on a high cost from at least 5$ Network Fee.

But if I would send the 1ETH over the Optimism L2 Chain, it promises me that I only pay around 0.3$ Fee, but I'm also open to a little risk that my transaction get's rolledup wrong.

So far so good.

Despite the Risk I go with the L2 Solution Optimism and send my ETH there. Here I have 2 fundamental Questions

1. Is my ETH after a successfull transaction from Wallet A to Wallet B still on the ETH mainnet as it would have been when sending it over the L1 mainnet or did I got some shitty Optimism Coins in a OP Wallet?
2. What signs the Transaction on the OP and ETH Chain, is it the L1 ETH Wallet which signs both Transactions or do I need to sign the ETH to OP Transaction with L1 ETH and than the OP to ETH Transaction with OP?

Yes it might look a bit confusing if you are a insider, but this is one thing I didn't get so far and therefore stayed away from L2 Solutions. Because I follow the Idea: "Only use, what you understand".