My thinking is, "Proof of Stake" is just a name for a set of rules which allow exchange funds and prevent double spending without expensive mining involved an "Stake" is not necessarily the amount of funds.
No. In proof of stake blocks are forged by a set of validators who have a locked amount of funds at stake to incentivize good behaviour
So, from this point of view, LN is a rudimentary PoS system where double spending is controlled by all the participants
Not at all.
There are no blocks to be produced no transactions to be validated, no native coins to be produced, there is no stake locked up to validate transactions.
The lightning network relegates double spending protection and all that to the blockchain
The only resemblance with a proof of stake system is the penalty for attempting to cheat the system by broadcasting a stale channel state.
Again false.
Sidechains are their own blockchains while the Lightning Network is not a blockchain but a "layer" on the bitcoin blockchain.
Infact you can deploy a lightning network on most blockchains that support HLTCs.
If you call any offchain payment mechanism a sidechain then that means you also consider exchanges as sidechains too.