Can I throw away contents of my Steam Inventory?

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In the instance of your "No longer valid" gifts, I would suggest contacting Steam support if the "Permanently delete gift..."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.

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.