If I use Arwen Undomiel's ability to discard a card and add a resource can I also trigger Elven Spear off the same discarded card.

No. The cost of an action are the words that come before the word "to". Both actions have the cost of discarding a card, and you cannot apply a single cost to multiple actions.

If Elven Spear were worded "Response: After a card is discarded from your hand, attached character gets +1 attack" then you could trigger both since the Spear no longer requires a cost of discarding a card. But it would completely change the Elven Spear card because you could no longer trigger it at will.

No. The discarding for Arwen Undomiel and Elven Spear are both costs , so must be paid every time you want to use the action. For all actions of the form "Do X to do Y", the X is the cost and must be paid specifically for doing that action. It's a bit like paying the resource costs for cards: you can't pay the cost for Squire of the Citadel and then use the same resource to also pay for Snowbourn Scout.

If Elven Spear would allow you to use another effect's discard to trigger the ability, it would say "Response: When you discard a card, blah blah blah".

Edit: lol.

Edit 2: For completeness, a card that allowed you to discard a card to boost attack like Elven Spear and also triggered off other discarded cards would probably be written:

Action: Discard a card from your hand.

Response: After a card is discarded from your hand, attached character gets +1 attack until the end of the phase. Limit 3 times per phase.

This would be a rather powerful card.

I guessed that was the way it was supposed to be but I though I had better ask (for whenever the heck the UK gets hero Arwen).