Is a new snitch used everytime for a new game of Quidditch?

Question

The snitch manufacturer wears gloves so as not to handle it directly. Scrimgeour brings the first snitch that Harry ever caught because he hopes to use its flesh memory for it to identify Harry. This implies the same snitch cannot be used between games since it will probably remember the previous person who caught it.

So,

1. Is it necessary to use a completely new snitch for every game of Quidditch?
2. Can we erase the previous memory of a snitch for the next game?
3. If the answer to 1 is yes, did Dumbledore really hold on to the first snitch Harry ever caught for all those years just to use it as a box for the Resurrection Stone which he did not even know if it existed at the time?
4. If the answer to 2 is yes, it is probably something Madam Hooch would do, may be after the current game ended or before the start of the next game. Why would she hold on to the snitch and not erase its memory?

Answer 1

1. If we assume that there’s no way to wipe the flesh memory (see comments/question 2), then yes, each game requires a new Snitch. In Deathly Hallows, Scrimgeour explains the purpose of flesh memories to the trio:

   “A Snitch is not touched by bare skin before it is released, not even by the maker, who wears gloves. It carries an enchantment by which it can identify the first human to lay hands upon it, in case of a disputed capture.”

   If you were using the same Snitch between games, especially in competitive or professional games, then you could have a problem with reusing Snitches.

2. As per comments: if this were true, then why wouldn’t the manufacturer just wipe the flesh memory blank?

   Making the flesh memory permanent serves as a security feature: it means it can’t be wiped if there’s a dispute, and acts as a consistent and reliable testimony when ruling on the outcome.

3. It's hard to say. Sometimes the players keep the Snitch that they caught. For example, in Order of the Phoenix, after the first Gryffindor-Slytherin match:

   The Snitch he [Harry] had caught earlier was now zooming around and around the common room; people were watching its progress as though hypnotized and Crookshanks was leaping from chair to chair, trying to catch it.

   On the other hand, when Harry caught his first Snitch, he was unaccustomed to the after parties and festivities surrounding a game: in the rush of the moment, it would have been quite easy for the Snitch to get lost (read: taken by Dumbledore).

   I don’t know how much foresight Dumbledore has, but the fact that Harry caught the Snitch in his mouth and not with his hand made it fairly unusual. Perhaps Dumbledore put it to one side as something that “might be useful later”, as anybody who wasn’t at the game wouldn’t know this useful detail.

   Indeed, even somebody who was there might easily forget:

   “The Snitch I caught in my first ever Quidditch match?” said Harry. “Don’t you remember?”

   Hermione looked simply bemused. Ron, however, gasped, pointing frantically from Harry to the Snitch and back again until he found his voice.

   “That was the one you nearly swallowed!”

   It depends on how much you trust Dumbledore. Personally, I think it’s quite plausible that at some point, he found the Snitch and kept it safe. After all, the used Snitches have to end up somewhere. Why not his office?

If you’re interested in the minutiae of Quidditch balls and rules, then there’s quite a good book called Quidditch Through the Ages by JKR which explains a lot of this stuff.