2) Smart people here have already answered: "It's impossible."
My question concerned only the technical possibility of sending this small amount.
I'll just give the technical answer you need, it's possible despite being difficult and considered not practical. You'll need to,
1. Find mining pool that accept such transaction. For example, MARA Slipstream.
2. Find wallet which let you create transaction where it send very small amount of Bitcoin to other address.
3. Create the transaction and get it's raw transaction.
4. Submit the raw transaction to the mining pool.