Steam – Activating a steam gift from Russia failed. Now it says the gift would already have been used

This didn't use to be possible, but it has been fixed now. The old gifting system has been integrated with the inventory.

When you click the "Accept Gift" button in the "pending gifts" section of your inventory, you'll get the option of storing it in your Steam Inventory.

I tried sending myself a gift just now - the email contains the sender's email address, indeed. What you can do is send a gift to your own secondary email address, and send them the URL for redeeming the gift (the URL you receive by emailing the gift to yourself) . Hopefully you have some way of contacting them other than their steam profile; for example, give them the url written on a slip of paper.

Caution: The URL contains the email you sent the gift to. If you don't want the receiver to be able to guess who you are, and the secondary email you're sending to is obviously related to you, create a new email account to send to. Example URL, email highlighted:

https://store.steampowered.com/account/ackgift/1F631467BB7E?redeemer=Fadeway%40yahoo.com

With this method, the first time they learn the sender's name will be when they enter the URL into a browser:

If there's no way to transmit the URL to them without posting on their steam profile, that's a lost cause, unless you get a third person to act as an arbiter/organizer who friends everybody and distributes the links while keeping it a secret who sent which one.