Apple alias in Mail and iCloud revealing my identity
I am unable to replicate this behaviour – when I click "Reply" in the native OS X Mail app, having All Inboxes selected, the address and correct alias name pops up. (If I only have the iCloud-inbox selected, it automatically replies from the main iCloud address..)

One thing you should do is double check the alias' settings on iCloud.com – specifically the alias name. If it's filled correctly, try to remove the iCloud (mail)account from your mac and relink it. If that doesn't fix it, go all the way and remove the actual alias from iCloud and make a new one, and try again. If even that doesn't fix it, throw your mac on the ground and walk away angrily. (No, don't actually try that, it won't work)

If nothing works, and you are unable to replicate this behaviour on a different computer, delete all settings and completely reset the Mail app.