IIRC Counterparty burned bitcoins for the initial distribution of their XCP token, maybe they've burned more coins since then but that's beside the point. When I refer to OMNI and Counterparty creating a currency of their own I don't mean their native tokens (if you can call them that) but I mean the tokens that are created using those native tokens. USDT in the case of OMNI.... and... sigh... Rare Pepes in the case of Counterparty, for example. The denomination, amount, value etc. of these tokens being largely independent of the value of the Bitcoin transactions underneath.
Like I said above: I think the clearest distinction is that sidechains run on separate blockchains while layer 2 solutions in general not necessarily require one.
Or another view: Layer 2 can be seen like the layers of the OSI model, eg. HTTP running on top of TCP/IP (I believe that this is even the root of the L2 nomenclature in the first place?) while sidechains connected to a main chain can be compared to LANs connected to the Internet. Fun thing is this analogy even accounts for LANs requiring protocol layers on top of the base layer to function, just as sidechains need more than just basic transactions.
I believe what you are describing is pretty much what is described in the channel factory proposal for a potential layer between LN and the base layer:
https://tik-old.ee.ethz.ch/file//a20a865ce40d40c8f942cf206a7cba96/Scalable_Funding_Of_Blockchain_Micropayment_Networks%20(1).pdf
Apart from that I believe you can still add another layer on top of LN in the form of colored coins via LN? Though the viability of that seems to be controversial.
No, atomic swaps take place across blockchains.