Is there any way to play online without putting the mod on the public list? Just me and my friend on the server, is there any way I can send him the mod and play without the mod on the list?

Translated by Google Translate

Yes, mods have a public and custom access option. You can choose custom access on a mod you own and type his username in the field. This way it will only show up for you and your friend. But you have to upload it and it has to pass a review process for obvious copyright reasons.