How does muun lightning wallet requires no trust and no backups?
I'm the CEO at Muun wallet.
Muun does rely on backups since the Emergency Kit is a backup, just not using mnemonics. The problem with mnemonics is that they don't contain a lot of stuff needed for the recovery of a multisig / lightning wallet, like the derivation paths, scripts, etc.
When using Muun, you will back up two pieces of data: the Recovery Code and the Emergency Kit. The Recovery Code is a randomly generated code that you write on a piece of paper, in the same way that you usually do with mnemonics. In fact, this code has higher entropy than a 12-word mnemonic. The Emergency Kit is a PDF document with all the stuff needed for recovery but harmless by itself. It has:

The two private keys (Muun is a 2-of-2 multisig), encrypted using the recovery code.
The output descriptors, with the derivation paths and scripts.
Instructions on how to use this data to recover your funds.
All this data is also available inside the PDF in a machine-readable format.

When using the recovery tool (https://github.com/muun/recovery), you will be able to move the funds using your Recovery Code and Emergency Kit combo. The next version of the recovery tool will use the machine-readable metadata in the PDF for easier input and better UX.
Regarding the lightning aspect of the recovery process, Muun uses super optimized submarine swaps to execute the lightning payments, both incoming and outgoing. This means that your funds are on-chain and thus recoverable by the mechanism I just described.
From the next version, Muun will start maintaining off-chain funds, and thus it will need to backup the off-chain commitment transactions. These transactions will be included in the PDF metadata. I'll update here with more information once we release this.