Can I throw away contents of my Steam Inventory?

I forget where I heard about it and I haven't used it myself, but TrashBot claims to be happy to take those one-sided trades, so you can use it to dispose of undeletable items. For as long as it's still running, anyway.

Yes you can for certain items - for example, when I look in my Steam inventory at my many copies of Dota 2, I can select one and a "Permanently delete gift..." button appears as per below;

This option doesn't appear for all items, for example if I select any of the new trading cards they do not have a delete option and the option is not present for the copy of XCOM I have from the Tomb Raider pre-order - in these instances you can gift them to your friends or, if they're marketable, sell them on the Steam market.

In the instance of your "No longer valid" gifts, I would suggest contacting Steam support if the "Permanently delete gift..." option is not available for you.

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According to the group's overview:

What we wish to do with the Coupon Dumpster is to make it easier on everyone. By providing the service of collecting, listing and finally giving the coupons away to you, the community.
Here, you can dump your unwanted coupons on us, and we will in return give those coupons FOR FREE to others who want them.