Satoshi said that after all 21 millions was produced the miners would earn the transactions commissions, what transaction comissions if now the only solution for bitcoin scaling is the LN ?
LN (and sidechains, too) cannot "live" without on-chain transactions - LN needs them to open/close channels, and sidechains to burn/unburn coins transacted between different sidechains.
Also, on-chain transactions have always advantages versus second-layer ones: they're totally trustless, no attacks like "channel rewinding" or "weak sidechain becomes attacked by miners" are possible. The main disadvantage, the confirmation time, is irrelevant for large transactions.
In a scenario where really "everybody is using LN", on-chain transactions could become very cheap and no transaction fee would be needed. As on-chain tx do have advantages over off-chain tx, there will be people using them (apart from channel-opening/closing transactions).
In short: if the on-chain transactions are cheap, blocks will fill again fast, and transaction fees will rise. So the transaction fee should self-regulate and be greater than zero. In a massively used Bitcoin environment miners will still have significant income.
I guess what I don't understand yet is how do you validate a transaction who's inputs come from a deleted block? I guess maybe you just assume it is valid because the transaction is so old. That's an interesting concept. Normally, you base validity on the genesis block, but in this case you assume validity based on the current block.
Many proof-of-stake coins do something very similar - they use a checkpoint system where reorgs longer than X are simply not allowed. If X=1000, then the block 1000 blocks before the current block becomes the earliest block from which a "tree" of different chains is allowed to exist, and all blocks before are considered "finalized". Chains that depart from the tree earlier are ignored by clients. So transactions that are confirmed in blocks before this last "finalized" block are assumed to be valid (they're also finalized).
The same thing would occur in cfcbitman's system. However, I think an account-based system like Ethereum's (instead of an UTXO-based one like in Bitcoin) would be advantageous.
There is also an issue of determining the longest chain, since every chain will have the same length.
I think not - afaik the chain length would be determined by the accumulated difficulty since the last block which is considered "finalized". (Maybe I understand your issue wrong)