Questions about layer 2s and gas (yes, retail signal)

Say I send an ERC-20 token to my main-net wallet. If I don't have any eth, I cannot swap said erc-20 to a stable coin in any way whatsoever?

A lot of people talk about holding ERC-20 on L2s.

Are all ERC-20 tokens compatible with Layer 2s?

For example, I can buy IMX on coinbase or Paal, etc and send to Arbitrum network. From there, I can swap to a stablecoin? I couldn't find some when looking through swapping pairs on some DEXs.

My understanding is that while you can send an ERC-20 to arbitrum network, you cannot swap. You would first need to bridge. This means if you want to send $$$ to a L2 for lower fees, you have to buy a stable on CEX, send that stable to Arbitrum wallet, bridge from main-net to aribitrum, then finally swap for your L2 token(IMX, PAAL, etc).

To return to $$$, you would need to reverse all steps? Or, can you just send stablecoin from Arbitrum net to CEX?