Normally, for Normal Trap Cards and Counter Trap Cards, after they are activated and resolve they are destroyed and sent to the Graveyard. Continuous Trap Cards with the "Continuous" icon and Trap Cards like "Metalmorph" and "Blast with Chain" which equip themselves to monsters remain face-up on the field after activation.
These TCG rulings were issued by Upper Deck Entertainment and have since been deemed unofficial by Konami. They were previously considered official and can still be correct, as long as they follow the official gameplay rules, or unless Konami has issued a ruling that says otherwise. References: 
If you have two copies of "Magical Thorn" active on the field, your opponent will take 1000 points of damage whenever they discard a card. If you have three copies, they will take 1500 points of damage.
The effect of "Magical Thorn" is applied when the opponent discards during the End Phase to make their hand size 6, discarding for the cost of cards such as "Tribute to the Doomed", "Cost Down", or "Judgment of Anubis", and discarding for card effects such as "Delinquent Duo", "Confiscation", "Robbin’ Goblin", "Morphing Jar" and "Mefist the Infernal General".
The effect of "Magical Thorn" is NOT applied when you send cards from your hand to the Graveyard for things such as a Tribute for a Ritual Spell Card, a Fusion Material Monster, or the effects of cards like "Chosen One" or "Gravekeeper’s Watcher".
The effect of "Magical Thorn" is NOT applied when cards in your hand are destroyed by "Chain Destruction" or when cards are sent from the Deck to Graveyard by the effect of "Painful Choice". It is also NOT applied when cards in your hand are removed from play by the effect of "Lightforce Sword".
The effect of "Magical Thorn" is only applied when a card is discarded from your opponent’s hand to your opponent’s Graveyard. So if they have your card in your hand by the effect of "Exchange" or "Graverobber" and they discard it, they do not take any damage from the effect of "Magical Thorn".
If the opponent discards from their hand as a cost to activate a card like "Cost Down" and you chain "Magical Thorn" they will not take any damage. If your opponent activates a card where discarding is part of the effect of the card, such as "Card Destruction", and you chain "Magical Thorn", they WILL take damage because the effect of "Magical Thorn" will be active before the discarding effect resolves.
"Magical Thorn" deals damage per card, so if the opponent activates "Graceful Charity" they will take 1000 points of damage when they discard 2 cards. If they discard 5 cards to activate "Final Destiny" they take 2500 points of damage.
Even though "Magical Thorn" does damage per card discarded, it deals the damage as one lump sum for effects like "Dark Room of Nightmare".
"Necrovalley" does not negate the effect of "Magical Thorn".
The effect of "Magical Thorn" inflicts damage after an effect resolves. So if the effect of "Morphing Jar" is activated, the damage will be dealt after the new cards are drawn. This means if your opponent has 100 Life Points, and you have "Magical Thorn" active on the field but only one card left in your Deck, you will lose the Duel before the effect of "Magical Thorn" can deal damage to reduce your opponent’s Life Points to 0.
"Magical Thorn" will deal damage between steps in a chain. For example, both you and your opponent have 100 Life Points remaining, your opponent has two cards in their hand and you have "Magical Thorn" active on the field and a Set "Mind Crush". It is your opponent’s turn and they activate "Final Flame" to deal 600 points of damage to you, and you chain "Mind Crush" calling the "Sinister Serpent" you know they have in their hand. After Step 2 ("Mind Crush") resolves, the effect of "Magical Thorn" deals 500 points of damage to your opponent and you win the Duel before Step 1, "Final Flame", resolves.
Necrovalley: "Necrovalley" will NOT negate effects that are activated when a card is sent TO the Graveyard or that activate IN the Graveyard or that count cards in the Graveyard. So it will NOT negate the effects of..."Magical Thorn".
↑Konami FAQ: On the victory condition of "Exodia" occurring at the same time as the victory condition of a Player's Life Points reaching 0
↑ 2.02.1Konami FAQ: [Basic Rules] When activating a Trap Card, does that Trap Card remain on the field?